By remove, you don't simply mean {snipsnipsnipyank} right? Don't those wires need to be reconnected/rerouted somehow?
No snipping. Just unplug the TSCU from the harness and you are done. Of course, you might want to protect the harness connectors in some way from water entry. I suggest just getting a new/used TSCU if you find out that was the problem.
If the system still doesn't flash with the TSCU removed then the problem is in the flasher relay, or hand control. The hand controls are delicate on the inside so it's possible that something is sticking. Don't be fooled by the sliding turn signal button not centering - it never does unless you do so manually. But if you slide the thumb button to the side all the way it should spring back a small amount. Let me know if you need more info in this regard.