JWT: A solution to let the token expire after a certain time of inactivity?
JWTs are self-describing integrity checked tokens. They are not designed for the use-case you described. JWTs cannot be expired on demand, nor can their validity be extended.
What you can do with these tokens is issue new tokens, just like you described. This will not invalidate the old ones. You will end up generating lots of tokens which will expire by themselves.
Am I missing a major flaw with this approach?
JWTs are not designed for a full-blown session management. They come with various tradeoffs. One of these is the inability to update them or expire them on-demand. The workaround you described works perfectly. I wrote a short post about session management, which should help you decide whether to make the tradeoffs.
In my opinion, you should probably move to the "classical" session management model. It supports idle timeouts and you also won't end up with lots of "sessions" (tokens).