Yes, it does make sense and yes, I agree 100%! Those cognitive process tests have a lot to answer for (but not as much as Socionics - don't get me started there!).
A person's true MBTI type reveals precisely how their team of eight functions is designed to work together (as you so elegantly put it). In fact that's all it really does: it says nothing about the level of development of any of those functions, nor does it take into account individual life circumstances which may force people to use their less-preferred functions instead of their natural strengths.
There are eight team roles which can be filled by any of the eight type functions. Each function is expressed differently according to its team role. So instead of one description for, say, Si, we really need eight descriptions: One for Si as dominant, one for auxiliary Si, one for tertiary Si, etc.
There are only sixteen possible ways in which the type functions can combine to form these teams, which correspond to the 16 MBTI types.
This is why there are so many contradictory descriptions and so much confusion around the nature of the type functions, sufficient to cause many people to conclude that they do not even exist. Every MBTI type will have a different experience of each function according to the role it plays in their personal team.
8 functions x 8 roles means there are 64 possible type/role combinations!
What we really need is a good, comprehensive and accurate description of these 64 combinations. The only one I know of is a very brief summary by a person who attended one of Dr John Beebe's seminars. It can be found here:
Symbol Thinking: Mapping Jungian Archetypes on Cognitive Processes
It is too brief to be of much use, but it should be enough to serve as a starting point for compiling a more comprehensive and useful list. We have many knowledgable type experts on this forum who subscribe to this extension of Jung's original theory of types. Perhaps we should put our heads together and start doing this? It's a massive task and likely to be the subject of much controversy and debate, but IMHO it is long overdue.