As for the coaching from the stands perspective, it is probably one of my bigger regrets. I was always encouraging if I chirped anything "way to battle, kid". That's OK when I'm coaching from the 3rd base box. But not in the stands. It's so hard to stay quiet but it's best for all. My biggest regret was replaying the game on the way home. Great job backing up the play at 3rd - way to be alert. But then I'd say, "the 2nd time you were up with that 2-0 count, were you sitting on a fastball in a certain spot?". Kids hate reliving games like that and they dread the conversation. Wish I would have picked up on that earlier. If you have kids playing take it from me - don't say a thing after a game. If you want to teach/correct wait until the next practice.
I think there is a happy medium and it is very kid dependent. Really the best time to go over situations/errors/proper plays/etc is right when they happen. It's freshest in the mind and can be the best time to get it to "stick". However, many kids will tune out as you are stating so it ends up being counterproductive. Waiting too long and the kids forget the situation and it doesn't have the same staying power. It is a fine line and one where you have to take queues and approach it accordingly.
After a big win when there was a lot of good, my son would engage about the few things he could have done better and received the "good jobs" as well. The 2-0 swing at an off speed pitch. The forgetting to back up third on a base hit to right kind of stuff. After a good game he was all about trying to get better and go over those things. After a tough loss, don't even talk about the good things. Don't try to point out the positives. Just let him be and when he was ready to talk about it then we did (it took me a long time to figure this one out though as you want to console/help/etc).
Every kid is different and it's tough to not just do what you want to do because you want to help. I still don't agree with the stands encouragement during the game stuff. Direct coaching or having your kid look at you for instruction every pitch or play is terrible. Cheering, encouragement, positives are all fine. It's a game afterall. I will say as my kid has gotten older and has developed the ability to correct things etc when they are noticed I have yelled out key words that will help him get on track if he has lost mechanics etc. It's very few and far between and is only done if he is repeatedly missing his mechanics. It might be something like "front arm" or "stay back" and that's it. They are queues that mean something to him for a specific deficiency. it's not constant and I can understand how this is frowned upon and try and refrain as much as possible but as someone that has coached him on the field for 12 years it's hard to refrain sometimes.
Yea, I don't know about the not cheering. I think kids WANT that. They want nothing more than their parents excited for them when they make a play.
In basketball, that is the hardest for me. For my sons... it won't matter because I am their coach at least until they get to HS. For my daughter... I try to keep to cheering. Her teammates called me their "number one fan" because I would cheer not just my daughter but the whole team. Giving each a high five... well, all except the short one which I gave a low five to.... after the games. My daughter never looks to me for guidance.
As a coach though, our last game, I have two kids on the team that are significantly below average skill level. One of them got the ball near top of the key. This kid struggles to not air ball when he is at the free throw line (and he actually is always past the line as he creeps up) and I hear his father yell "shoot!" as he is nearly at three point distance and still struggling to gather the ball with the defenders about to be on top of him. As a coach, you don't want to hear that. First, it doesn't do the kid any good and second if he is listening to that he isn't listening to me which I have spent a lot of time drilling into the kids that they need to know their range and stick to it. Open shots are fine to take as long as it is in their range. This was so far outside of this kids range he could launch it 100 times and maybe hit something once and again... he didn't even have control of the ball when he gets shouted at to shoot. sigh.