You acknowledge that America “has an interest in a cooperative relationship with Beijing,” yet you defend an action that undermines that interest. It might have been worth it if the United States got something in return, but America got nothing.
The aspect you’re leaving out is domestic audiences. America officially maintaining the “one China” fiction while helping Taiwan defend itself gets the U.S. the best of both worlds. Help a friend maintain de facto independence (which is a thorn in the side of Great Power rival China), while giving the Chinese government a way to save face domestically.
Remove that veneer and the Chinese government will cooperate less with the United States. America hawks will gain power within the CCP and/or Xi will take a more hawkish line to avoid looking weak in front of his domestic audience.
I go into this in more detail here, if you’re curious: