I’ve heard Karl Keating address this “Q source theory” more than once on CAL. He says one of the big problems with it is the utter lack of mention of it in early Church history.
That is not a convincing argument, not at all
“The Case Against Q”, by Mark Goodacre, is a good book on this issue. He uses the same critical methods as those who are not convinced by the case made against it -
that is how it should be examined. Disagreements of this kind do not destroy Bible criticism - they are one of the modes by which it makes progress.
Even if untenable, the hypothesis has certainly been stimulating & fruitful, not least because it has encouraged the ever-closer study of each of the gospels; & this has greatly benefited appreciation of the gospels, both as literary works & as examples of early Christian theology. That has in turn benefited the study of the other NT books & their literary structure, theologies, & the like. And the better we understand the NT literature in those ways (as in others), the better equipped we are to appreciate the relations between the Testaments. Which in turn deepens the Church’s understanding of the relation between the Bible & the other means of grace, as well as of the relation between faith, theology, dogma, tradition & history. And that
in turn contributes to a better understanding of ecclesiology, the shape of the Liturgy, & of the Church’s mission.

These disciplines condition, assist, & strengthen each other. (And none of them is to be found in the Church alone, just as the sciences are not.)
The case against this or that element in critical scholarship - such as the Two-Source Theory, or the Four-Source Hypothesis (aka JEPD) - is (FWIW) not the same thing as arguing against critical scholarship. So even if Q (for example) should turn out to have less explanatory power than at present is supposed, the critical approach to the Holy Bible would no more be damaged (let alone destroyed) than physics has been destroyed by having to reject the phlogiston theory, or than medicine has been destroyed by having to reject the four humours theory.
It’s not clear that people always realise this

- & it would be a pity, if those who have concluded to their own satisfaction that the critical scholarship of the Bible is (even in principle) a bad thing, were under the impression that the discrediting of influential & widespread hypotheses meant the end of critical scholarship. An hypothesis which is discredited can (in principle) always be revived, because scholarship is not static; the considerations that invalidate it are as liable to be invalidated as the hypothesis, so today’s exploded hypothesis may in due course be re-established on more solid foundations.