How memories are stored certainly matters, it is too much of an assumption that levels are sealed off. Such an assumption may be implicitly negated in a model, but obviously this doesn't mean something has changed; the nature of material systems has this issue, unlike mathematical ones.

Another poignant property of material systems is that at times there is a special status of observer for them. In the case of the mind, you have the consciousness of the person and while certainly it can be juxtaposed to other instances of it, it is a different relation from the one which would allow carefree use of the term "anecdote". Notice "special", which in no way means infallible or anything of such a class, but it does connote a qualitative difference: apart from other means of observation - those available to everyone else, like the tool you mentioned - there is also the sense through consciousness itself, which here was for reasons of brevity referred to as intuition. 

Of course consciousness itself is problematic as an observer. But it is used - in a different capacity - in all other input procedures, since you need an observer to take those in as well. If one treats consciousness as a block which acts with built-in biases, it is too much to believe those are cancelled if one simply uses it as an observer of another type of input. It's due to this (particular) loop that posing a question about intuition is not without merit.