What examples do we have of mathematicians who explicitly and publicly expressed their personal confidence that mainstream modern logic, as used in mathematics, either as object of study in itself or simply as a tool, was an appropriate representation of the sense of logic most of us have without having to study formal logic?
Nobody is going to claim that things like 'Ex Falso Quodlibet' or the Zermelo-Frankel constructions represent natural human logic. They are formal dodges that avoid confusing aspects of naive logic on purpose.
One important example: The idea that you cannot have a set of all sets is not reasonable to most humans, naively. It has to be motivated by a need to evade paradoxes, and they eventually accept it, but it clearly contradicts a very natural impulse.
We go so far as to have different set-theories (e.g. Zermelo-Frankel and Godel-Bernays-von-Neumann) that do or do not allow for a universal set, because not having one seems too counter-intuitive to some mathematicians. In the latter, you can have sets that include all the sets, but you still can't have a set of all sets, because Russel's paradox still can't be permitted.
So the already artificial notion of 'too big a collection to be contained', the closest intuition we can usually impart for why there should not be such a set, actually fails to capture what is going on. There is a real gap here between the formalized solution and our vocabulary that humans don't actually seem to be able to accommodate fully.
But in the end, the purpose of formalization is to improve the system in some way. If it captured all the confusing parts and all the potential paradoxes, it would not actually achieve anything.
Automated theorem provers, such as Otter, or Prover9 usually use a subset of first-order logic. There have existed open mathematical conjectures which first got solved by theorem provers, such as the Robbins problem. There are some mathematicians, such as Ken Kunnen, who use theorem provers extensively in their work also. So, I think the answer to your question is 'yes'.