Experience should be added among player skills, increase by the amount of games played using the same criteria as in training. 

It should also be use as the bigger factor in picking captain 

Is this not exactly how it works at the moment?

Experience determines best captain

Best captain gives bonus to team

Experience is gained by playing matches

Realize I didn't explain this properly.

Currently you have to visit each player one by one to see experience, in skills you can organize players by stat to see who have the highest and lowest in each category.

Not all games are used now, friendlies are not 

Don't feel it is the main factor to determine captain now, my most experienced players are not the best captain 

DId you check the Player roles option in the tactic screen? You can see the best option for captain there.

Indeed. Players "experience" is kidna useless at this moment.