Can a tradition question itself?

Yes, and it has done so for a very long time. Classical Indian philosophy grew through debate. A thinker was expected to set out the strongest version of the opposing view, the pūrvapakṣa, before answering it. Schools within the tradition argued with each other, and with Buddhist and Jain thinkers, for centuries.

Questioning also runs through the texts themselves. The Upanishads are full of students pressing their teachers. In the Katha Upanishad, a boy turns down every reward Death offers him until Death answers his question.

So perhaps the better question is not whether a tradition can question itself, but who gets to ask, and whether we listen to the answers.