At a grass roots level, first aiders (if they are really present) are barely more skilled than anyone else present - note that I am aware that some teams may have medically trained staff playing or spectating which is a different scenario. The 3 day 1st aid course (which I have done in the past) gives you minimal knowledge of dealing with minor injuries and what to do for some more dangerous conditions (boils down to get an ambulance and basic life support). While I would listen if they felt a player could not be moved following an injury or something of that nature, if I am told a player can continue but I believed it would be dangerous, the player does not continue to play even if that means game suspended/abandoned. Law 5. That said, no real first aider/physio is going to allow a player to continue if he is so obviously endangered that I felt I had to step in.
I am not even sure if grass roots teams have to have a first aider who has done the 3 days course? Don't they have to do the 1 afternoon first aid for football course, no evaluation at the end? I have done that to. It does not make you a physio.