The majority of clubs will have a physio or person with first aid experience in the dugout on match days, and possibly training nights twice a week, paid or voluntary this will be on a par time basis. I can't imagine junior football clubs having the funding or the need for a full time physio, as demand and player availability is limited
Individual players may want to obtain the services of a physio outwith the football club, that is their choice, but they may need to pay for this privately, or through injury insurance this will probably be when the injury is more severe and rehabilitation will take longer for that individual to regain their fitness.
the player you mentioned that was injured and left to recover on his own will be i'm sure an isolated occurrence, and not the norm within the junior game.
As of those physios that do the job voluntary, will in most cases be supporters of the club and understand the financial restraints and well being of the players within the club,
For me personally I love being involved within my club doing what i can in some small way to help