I think the simplest explanation is the correct one: they can't summon a sane Garuda, because they simply can't envision her that way. Their prayers are to Garuda, so they take the form of the Garuda they have always known. And then once tempering happens, well...

Now, if there was a sect with an oral tradition of a sane, stalwart Lady of the Vortex, would that Garuda be sane and stalwart as well? And what would happen if that version encountered followers of the insane version? Would they be able to summon their version as well, or would they only empower the current summoned form in spite of its differences?

I think I'm getting closer to a way to justify FC summoning...