Maybe it's just me, but I think this is entirely untrue and exaggerated It didn't come into my mind as a way of strengthening whatever arguments I may have, but as a hypothetical possibility.
Well, I don't agree with the "possibly (probably?) leading to exactly the same result" bit of your original post, but I guess we're disagreeing about two different hypothetical outcomes of a hypothetical scenario, as you say. I do think though that it's possible in this case to assume the "If they hadn't written LPO we wouldn't have taken this action" line is not a lie, and make a case against what has actually been done (no doubt by a management acting under its own perceived pressures, which needs to be convinced that it should have acted differently, not told that we believe it would have acted in worse faith than it already has done in a different set of circumstances).
I don't really think it's productive to talk about "reasons why orchestral musicians might take a more mainstream political position", after all one could invoke such reasons about university professors and all sorts of other people too but they wouldn't always be true.