I actually don't have that much of a problem with people who say they're INFx, or ExFJ, etc etc. I think it's possible, especially when you get to individuals who are incredibly balanced between multiple functions.
But I also don't think all of the various theories tie together as seamlessly as people think they should. I think a blending of all the theories is reasonable. I think the dichotomy approach can be valid to a degree, because I think there are WAAYYYY too many factors with a pure cognitive theory approach that complicate things. A lot can depend on how one was raised, and ones' life experiences, in how adept one becomes at using various functions, as well as how someone might begin to 'flex' and consciously choose one function over another depending on the scenario. Maturity throws further complications into it.
I'm also not a big fan of my 'type' limiting my abilities, or using my 'type' as a crutch to say, 'Well, my dominant functions are XxYy, so that's what I use most of the time, that's just how I am, so I'm rarely going to use other functions, and all of my observable behavior is going to be the XxYy'. I'm not directing this at anyone, but that's the problem I start having when people emphasize the cognitive function approach more than other approaches.
Then you have the brain dominance theory too, but that doesn't work for people who are pretty much balanced between right and left (and I think I am).
When I take into account ALL of the theories, as well as my behaviors and traits growing up, and also account for conscious choices I make and things I have learned over time in interacting with others, and how I WANT to interact with others, I arrive at what I've decided is my best fit. Is it possible I'm another INxx?? Oh, sure it is. A few months ago, when I looked at pure cognitive functions, I could relate to Fe, Fi, Ne, and Ni very much. All of them. So I could be a chameleon, too, of INFJ, INFP, and INTJ (although I do think INTJ is out of the question because Te is totally weak for me, and I have little motivation to go Te on anyone). Would that then be a truer reflection of who I really am? A blend? But then that would go against the theory of everyone only having ONE real type.
To be honest, most INxx responses on this site resonate with me on one level or another, but INFx certainly more so, and INFJ the most.
All of this is to say -- I can relate to those who have an 'X' in their type, as to be honest for some I DON'T think it can be easy to determine type at all - esp. w/ the more balanced individuals or someone who has matured and often doesn't go with their 'kneejerk reaction' as much. I know for me I've pondered my type for years.
Further reflection and introspection on my personality type would be asinine, because much of my life IS introspecting and becoming more self-aware, so thinking about it more than I already do is totally pointless and wouldn't do a bit of good!

. And I would imagine that people who have 'X' or are uncertain of their type have spent an equal amount of time pondering all of it, and genuinely can't make the call. I don't think it's THAT easy for everyone - only those who are very dominant in certain functions.