Organizing advice?

I’m putting this here instead of parenting because those who aren’t parents may still have good tips.

I have a bunch of little sets of things like this for my daughter: (music, stacking cups, bowling, cooking, etc) - all little sets of things that I want to keep together.

But here’s what I store all her toys in:

I have no space for of those big things that go on the wall with a bunch of little shoe box sized plastic bins. I would just store things in gallon zip loc bags and toss them in the toy chest to keep them together, but bags are a suffocation hazard.

So, any thoughts on how to store all this stuff so the parts can stay together and not get lost/separated in the big toy chest?