This is the best post in the thread IMO and it is how this exercise makes most sense. It is much more useful to know how to construct complete winning team then just say that QB/DE/LT/whatever is most important. Thing is that you always have chance to have some elite players, some average and some clear weaknesses, and you have to realize this when constructing a roster. Good GM knows how to allocate resources. It is obviously more complicated than this, since you have 53 players on roster and not 24 (and having good CB3 might be for example more valuable than having good OLB2), but this is a good starting point.
As for the model itself, I'd slightly change naming structure - you can't have 5 All-Pro players in this exercise. I would also avoid "Barely NFL Caliber" term when talking about starters. I'd say it should go like Elite/All Pro (2x), Pro Bowl (3x), Solid/Good Starter (7x), Avg Starter (7x), Borderline Starter/Backup Level (3x), Liability (2x). As a base your distribution 2/3/7/7/3/2 looks reasonable to me.
I would also probably allow people to move players among groups, i.e. allow people to decide that they can upgrade one Liability to Borderline Starter in exchange for downgrading one Pro Bowl to Solid Starter etc.
I don't pretend to be an expert but my initial thoughts are as follows:
All Pros (2x): QB, LT
Pro Bowl (3x): DE, CB, WR
Good Starters (7x): C, RT, OG, MLB, DE2, DT, K (yes kicker here)
Avg Starters (7x): OG, OLB, WR2, RB, TE, S, CB2
Backup Level (3x): DT2, OLB2, P
Liability (2x): S, WR3