Okay. In your example. First someone casts the spell by holding their hands out. Flames shoot out. Then they have cast the spell and now can do anything else they want to do with their hands. You don't have to spend the whole turn casting the spell. You only spend an action. As the rules say, you use the Somatic component while casting the spell. Then anything else you do on your turn is after you have already done everything involved with spell casting. Its not rules-lawyering, its how it works. If the GM wants to make it more involved, that's fine, they're the GM, but that isn't how it is meant to work vanilla.
In addition, controlling the hand does not require S component, only casting the spell does. It says nothing about having to move your own hand to control it, or point at where you want to control it, or anything like that. It doesn't even say to point at where you want the Mage Hand to appear when you cast it. Nothing about the spell says anything about pointing.
What do you mean by keep concentrating, neither Burning Hands nor Mage Hand are concentration spells?
I'm pretty sure you are allowed to cast a cantrip and bonus action spell even if they both have S components.
Or are you saying you're rules-lawyering? In which case, I don't agree, nothing you're saying last post is in the rules.
Tangeant: If your holy symbol is on your shield, apparently you can use both Material and Somatic components with that shield, but Somatic spells without Material components need an empty hand. Weird.