HTT - this isn't me being critical, so read this...
The concept of picking numbers to determine teams is quite common and still used nowadays (I know thats how my 11-year old brother chooses teams up at the field) and it was used sometimes when I was at school. There wasn't a "formation" as you said but there were kids who defended, kids who went in net, kids who goal poached, kids who just followed the ball everywhere and kids who just wanted to try out that trick they saw Henry do on MOTD. Your about 85% right about the coaching thing, it does pretty much stink, personally I think a child under 10 should be played by a coach as a "keeper", "defender", "midfielder" or "striker", none of this left back, right back, attacking mid etc. They should be encouraged to use the space out wide and told/shown how it can pay off etc but not rigidly made to hug the touchline. They should get to have a go at every position (keeper optional) and decide where they enjoy playing or where they play best.
After that age of 10, the coach should start assessing where he thinks kids are best suited to play, and sort of having a chat with them and asking. Obviously a lot of kids will say "striker!" but the coach should still say "I don't think you'd be at your best as a striker, you need to have really really good finishing/heading whatever, and if you played midfield you'd be able to use your good pass and tackle as well as having some shots when you can." or things to that effect, and in my opinion, if a kid is told to play in defence but then has a few good games and is praised by the coach, they'll grow to the idea of playing there.
When I was playing for the school team in middle school, the first eight weeks or so was when I was on the bench and I only got on the pitch something like three times, for about 10 minutes at the end. I went to the teacher in training the next week and asked him if I could play more, and he said "Yes, if you keep it up in training you'll get more games." - Although I was benched again that week, I played the whole second half. I went to him and had a similar conversation the next week and he said he was pleased how bothered I was about not playing. After that started every game bar two (one I missed through illness, the other was the last game of the season and I was dropped to the bench) and sort of developed a consistency of playing at centre midfield. Through that, not only did I gain skills of how to mark other kids and learn to pass quicker, I also got the mental part right of not giving up and hassling the manager to play and trying to prove I was good enough.
Many a manager has picked me as a CB in the past also, as you said HTT, because of my height. The problem is, I was never a good enough reader of the game, or good enough at holding a defensive line/position to be any good there. I was always drifting about or playing people onside. Striker is where I'm better at holding a line, retaining the ball and trying to find a pass. There's also less chance that an error will lead to a goal, and as my passing isn't exactly great, it would be a disaster if I played at CB.
Kids need to enjoy football, but also be coached and given some form of discipline, getting slightly firmer at every youth level.
They also need to be aware that if they are very unhappy with where they are playing or that they aren't enjoying it, that they must speak to the coach politely and try and rectify the problem. Coaches need to make that more plain.
