Well, if you can determine your type, then cognitive functions can help you determine any imbalances you have, for example if you're overusing your inferior function. It suggests that there is a proper way to function based on your innate wiring and that there are ways to function correctly or incorrectly. But if you can't determine your type, then it's a bit useless, unless someone does it for you, but the information regarding the subject will probably become more concrete later on so that it becomes easier to do.
So I don't know if it has been hijacked or whatever, but it certainly requires being open to the possibility that you're imbalanced, and being willing to take steps to remedy those imbalances. It tells you what your baseline functioning roughly is and how you're deviating from that baseline.
I don't know whether I can say that the theory has "helped" me. I've more been focused on trying to help the theory by figuring out how it all works and what can be done with it. The theory is still in the phase of being effectively laid out so that it can be used, and it takes a long time at the present to be able to learn what's going on enough to see it.
I guess what it's done is told me my strengths and weaknesses. It's let me know that if I am constantly forgetting to do things, that that's just how I'm naturally wired as an INFJ, which is a type with one of the worst memories for concrete things. It explains that difficulties interacting with people might be due to typological differences in the sense that we place emphasis on opposite side of the information spectrum, and suggests ways that this can be dealt with and to what extent that it can. For example if I interact with an INTP there may be a conflict due to the differences between Fe and Ti, but if I switch the order of Fe and Ti in my cognitive function order then it may work better, or I may need to linger for a while when I get to Ti in the usual function ordering.
It gives one of the factors involved in personal relationships in other words, and can determine (alongside other factors that need to be taken into account) what kind of potential a relationship offers.
Another way that I've used it is for mapping out my timeline. I can see through the multi-type approach how there have been different threads operating throughout my life, like multiple stories all working together to created a unified story. If I'm perceptive enough, I can determine which storyline a particular behaviour is most likely coming from. It's a bit analytical in that it divides a person into lots of parts, so can have the effect like that saying about how when you dissect a flower it loses its beauty, so by trying to understand yourself by "dissecting" your mind up you can somewhat lose the way that the whole is more than the sum of its parts, but it may be useful to do so nonetheless. I couldn't say for sure whether that method is or isn't appropriate for understanding a person, it's just a method I've been trying to understand.
There's plenty more. I don't know what use the Big 5 has. It seems like an approximate version of MBTI dichotomies, which might explain some things better than the dichotomies of MBTI, but MBTI dichotomies would explain things that Big 5 doesn't. But Big 5 doesn't seem to have the idea of baseline functioning which can be deviated from, and it lacks the discrete structures that are useful for a more analytical approach. Have you found use for Big 5?
The problem with MBTI is that it can be tailored too easily to suit the person's biases & on top of that, very few people seem able to even agree on what the functions are or should be defined as.
I point to the myriad of threads, videos and general discussions on here and other platforms as evidence that cognitive function theory is very vague and unrealised. You may be able to justify that by saying you are figuring out the theory and how it all works as a way to help it and learn what can be done with it, but I just see it as a jumbled mess with an ever-increasing series of interpretations.
I've owned a great deal of books on the subject, from Gifts Differing, to Jung's own
Psychological Types and Lenore Thompson's personality manual, to name a few, and it only serves, from my perspective, to cause too many different baselines to arise.
For example you gave a hypothetical interaction between yourself as an INFJ and an INTP and then mentioned (as one angle of resolving it) switching your Fe and Ti in order to ease a conflict between yourselves. But the interpretations of both your behaviour and the INTP's is left to a great deal of assumption, both around what each of you think or yourselves and each other in the opposite perspective. The basis for claiming one type over another is left to the wind, it's very 'out there' and nebulous. And while I admit there are limitations to the extremes of empiricism, that doesn't mean it's a good idea to go with a science so soft it's practically imploding under the weight of it's own contradictions.
And if switching functions is possible when necessary, then how is that different from the usual interpretation of social interaction? Where one or another is adapting and changing themselves to suit the situation? If anything it's grounds for less confidence in type, not more. People have been doing that for thousands of years without the need for an MBTI framework on which to hang that understanding. It's also how most professional relationships survive.
The reason it seems so useful for analytical understanding is that it is concerning the social interaction realm of human society. There are hundreds, perhaps even millions, of different ways behaviour and intent can be interpreted. We've been trying to understand this for most of our existence and, as a species, we have been through a great deal of different ideas and systems for understanding one another. Social interaction is emotional, chemical, a bit mysterious and tends to go on at a very quick and unconscious level of the brain. That means MBTI fits nicely into the spectrum as a system vague and undefined enough for people to see it as a tool for understanding those similarly vague social behaviours. There's a lot of instinct and intuition in understanding other people's behaviours and motivations and we're really good at it, generally speaking, which is why we can usually read one another to a certain extent and predict some actions and moves.
Ironically, at the same time, the MBTI theories also tend to make the mistake of occasionally going too specific and defining certain tasks or job areas as most suitable to a certain function or type. By contrast (and you asked me how Big 5 is useful) the dimensions of the big 5 simply outline areas that you may or may not be suited to. For example if you are high in openness you may not particularly enjoy routine jobs with set activities, another example is those high in the trait of neuroticism may not be suited to activities of high pressure and responsibility (although it may have a converse affect and make people try harder under such pressures). Those are some very simplistic examples, for the sake of brevity, but it can easily be used in a more in-depth and analytical way. By contrast MBTI is simply piling complexity on top of complexity....and I normally like a degree of complexity, but it has to be a purposeful and useful complexity, not something that comes across like a social game, or the equivalent of people discussing their horoscopes from a newspaper, except taken to a ridiculous extreme and with seriousness added on top.
You say that MBTI has a baseline whereas Big 5 doesn't, I just don't see it, if anything the opposite is true. MBTI doesn't seem to know what it's baseline is, or else people keep picking and choosing the baseline that suits them the best. Big 5 on the other hand is more neutral and doesn't attempt to give a good or bad appraisal of your traits (with the exception of openness which is like big 5's version of being an INxx type when it comes to self-grandiosity
***).
Another issue with MBTI is the gathered data on job type correlation (and lets not brush over the fact this theory was essentially adapted from Jung's work for the purpose of business application). With job correlation I find it very convenient that people in certain jobs, give their type most usually as the type that is correlated with that job. For example, I have seen statistics (and there are many on here who have posted these) of those in scientific fields, most commonly give their types as INTJ or INTP. Which seems very suitable for MBTI, given that those types (in both cognitive functions and general descriptions) are correlated with scientifically based jobs. As I said: How very convenient for the theory. There is a term for that, it's a bit of a cliche, but it's known as a confirmation bias.
I don't know how many will have the patience to read this far, but please be assured I'm not just blindly hating or ragging on MBTI for trolling purposes or personal pleasure. In truth there is a part of me that wishes, I think, that it were actually more reliable and useful than it is, because then I could justify the time I spent on it and use it for the betterment of myself.
People are free to carry on with the theory, I won't stop them, but I also won't ignore any criticism and flaws within that system. If you can give it a fair appraisal and judge for yourself, well there's nothing wrong with that. I might be proven completely wrong about MBTI one day and I accept that as a possible outcome, but I don't see that happening any time soon.
Just don't bet all your fragile cognitive eggs on it.
*** Which isn't to say I believe INxx types (whatever or whoever they might be) are prone to self-grandiosity, but that the apparent 'rarity' and 'exceptional' nature of those types as most commonly defined, tend to attract people who want to be seen as a different or exceptional individual for their own emotional well-being and ego.