Many non-math fields produce research that is valuable, but with no real impact on math itself. This seems to suggest your guess/hypothesis is false. Some use math, but more use statistics. Some use neither.
A typical thesis in many fields is novel for the questions it asks, though the way of gathering evidence and the statistical evaluation of that question can be quite standard. Such theses don't advance statistics either, though they may (occasionally) use it in creative ways. But their importance lies in asking and giving evidence for important questions in the field.
But there are fields that use very little math. Some theses in philosophy, for example, don't use math at all. History might be similar.
But Economics is pretty mathematical in some of the things it does. But since it often gets things quite wrong, while using "accepted" methods, it isn't really fundamentally like math. You need a lot of math to study it, perhaps, but that doesn't make it math - certainly not pure math, though it is an application (many cases).
I'd guess that you probably do need to prove that you know some math in order to undertake a dissertation in some fields and likewise statistics in others. In particular, in some places there are comprehensive exams that require some math and/or statistics background.
But, it isn't a universal. It is field dependent and even likely dependent on some specifics within fields.
In the US, most undergraduates going on to graduate study will have some math required in their program. Political Science students, perhaps, not so much, though some questions there require statistics for adequate answers. English Literature maybe not needed at all even if it is studied.
Note that in most fields, questions don't have binary, yes/no, answers. So statistics is often necessary to get the measure of "how true" some hypothesis is. And, since it is difficult to study complete populations for many things (as some are large are others constantly changing) it is necessary to rely on (imperfect) sampling. But it is the questions, themselves, that drive the research.