Entropy just is, it's not trying anything. Same for nature.

Cells actively remove proteins which arem't needed anymore. E.g. Ubiquitin gets attached so the proteasome can recognise them and break them down. Protein decay happens over time as well but that's also a good thing for cells. It's part of the waste management to keep cells from getting stuffed with old proteins that can aggregate together.

So no, they are not destroyed faster than they can be built. There's a whole system in place to determine what needs to go when.