The purpose of modern frigates is to perform specialized missions away from the rest of the fleet,
without having to resort to large ships like destroyers. If you're going to build frigates the size of destroyers, then it'd have been better to just build more destroyers to begin with. The LCS ships, for all their many many faults, at least understood this part. That the main roles for smaller vessels are ASW, minesweeping, and the occasional surface combat (mainly to fend off smaller vessels). These roles are crucial, but you don't want to allocate destoyers to performing them because it's a waste of resources and because an isolated destroyer is far too vulnerable to attack.
FREMM and Constellation are designed to be able to fully defend themselves during independent operations, I don't see why new Chinese frigates will not have that feature. I say defending itself independently from antiship missiles and aircraft within a reasonable range is a bare minimum requirement. ASW operation within the 1IC can be literally done by anything, drones, 056s, etc. Frigates and above should be dedicated mostly to blue water operations deep in the 2IC and beyond.
I think that the ability to "fully defend themselves" is junk. No vessel by itself is going to be able to do that with any consistency, and insisting on this capability is designing ships wrong. Burkes can't do it, Ticonderogas can't do it, and there's no damned way that FREMMs or Constellations are going to be able to do it.
In any case, FREMMs are already too big to really work as frigates. A more limited navy has a use for a more general purpose frigate to act as mini-destroyers, but this isn't the case for the PLAN. And Constellations are just silly. By the time the design is finalized, it'll probably be a bigger ship than a 052! At that point, the USN should have just bought more Burkes and admit that they're just going to have to live with a 0 frigate navy.