Boxes are only useful after many years if you intend to sell complete sets. If you have kids playing with the sets, chances are the boxes won't be very useful even if you intended to sell the lego in future. I'd throw 'em out.
Once a set begins to break down (either gets played with and breaks, or kid loses interest, or uses part of the set to build something else), I'd break down the rest of the set completely and add the parts to a pile for them to play with.
As for sorting... if all your lego fits into one of those containers, then maybe you don't actually need to sort. Let chaos reign! Give your kids some direction - "can you build a house/spaceship/crocodile/alien?" and let them hunt for parts. Kids are usually more creative and build with what they have, so part discovery isn't a bad thing.
If you really want to sort and make it more organised, with the numbre of parts you have, I'd just sort by size (small/medium/big pieces). digging through a big pile of unsorted lego for that small 1x2 plate is probably the most annoying thing for a kid.
Additional reading: https://brickarchitect.com/guide/