IIRC we had two Flashes concurrently

Yeah, they were just both the Flash. Dick and Bruce were in the same situation at around the same time.

Wally was already in his mid-to-late 20s and was a father to two children when Barry returned. There’s no way the “Kid Flash” moniker would’ve fit hit anymore.

If memory serves me right they both operated as the Flash. Of course the New 52 erased our Wally and gave us a different one. Didn't care for that idea.

I get it, DC wanted to create a new Kid Flash. Did we really need one? Not really, but we have one.