There's frankly not enough pressure within the theory community to disregard cagey theorists that actively avoid testing their predictions, or disregard the importance of testing in general.
I think having testability as a qualification for theory might not serve us well.
It blows my mind how much we have deduced about black holes in 100+ years, yet it was only very recently that we could actually "see" one. Would hate to lose that.
It wasn't team Einstein's responsibility to propose a solution, his motive was a rebuttal.
The fact that it went unstated for 30 years, Einstein died in 1955, reflected poorly on team Heisenberg.
The issue I see, is how to continue to study such purely theoretical matters?
Why should everything have to be testable (and thus, potentially developed into a marketable product)?
It may not be science, but a lot of people still like to do that kind of theoretical work.
The whole point of testing theories is there can only be "one truth". If you let theories run rampant without testing you end up in logically inconsistent chaos.
All that said, theories should certainly be given time to grow when experimental validation is not immediately available. However, there's many in the community that take great liberty with that long leash to the detriment of physics progress.