Jay is still active. Barry has come back. Wally is doing his thing. Bart also came back. Jesse is still with JSA. Did I forget any active heroes who are or have been named Flash?
In addition, Wally was Kid Flash. Then Bart was Impulse. But now Wally’s daughter is Impulse. This is because Bart was Impulse but then became the Flash and died but came back and is now Kid Flash.
So, is the Flash family getting too big? My answer would have to be not yet. There’s always room for another super-speed hero with ties to the speed force.