I am aware the format is still experimental and this may already been considered:

Less rewarded places more prizes.

Reward 3 out of 10 , 5 out of 16 ,etc. 

given the risk of injury ..more appealing prizes will be more tempting for some of us , even if there is the risk to actually win nothing.

In your 5 ou of 16 example, it would leave 11 teams with deadly squat... i think it can be dissapoiting for a lot of people.

I was thinking about awarding 1 game share for the winner, for every tournament made by the system, 1 or 2 tournaments per day...

Fair point, to be fair.

I think everyone should have fun.

4 tournament tiers:
An example
1 - AQ between 40 - 44
2 - AQ between 45 - 49
3 - AQ between 50 - 54
4 - AQ between 55 - 59

This way everyone only participates once, this will really engage people to participate(Series D, C, B, A).

Update: Regarding the prize, I believe it will be a product package (batatorade/trail mix/football kit/medical/document), or a drop (young people/coach), or 1 share as a prize (it will be locked for sale for 60 days to prevent cheaters).

I don't see why the tournaments organized by the system should differ from those organized by the users. There is already the possibility for extra pot funding, it is more than enough, why award shares or products?
Even if the tournaments are scheduled, it would still be unfair for those who are left out. It is supposed to be a football manager game, not a "rush to click" game.

@maridiba

I believe you can create/organize private tournaments by the user, both free and through the sponsorship fund to distribute prizes.

And these "2 or 4 tournaments" will be generated by the system by default and the prize is fixed... the team manager can request participation in any tournament. :)

If winners got a gameshare it also must give to winners in history, not only now.