In order to address this question, I think it is important to first take a step back and examine with a critical eye something that we normally take for granted, namely the professionalization and compartmentalization of academic departments. It is common to think of "philosophy" as "that which is practiced by professional philosophers" and "mathematics" as "that which is practiced by professional mathematicians." However, in my opinion, the divvying up of academic "turf" is driven more by sociological and economic factors than by any intrinsic divisions in the intellectual subject matter. To put it crassly (and somewhat exaggeratedly), my fellow academicians and I stand to extract more money and prestige from the rest of society if we agree to slice up the pie in a certain way and not fight too much internally over the division, conserving our energy to be directed outwards.
But if we actually want to understand the true relationship between "philosophy" and "mathematics," we shouldn't confuse ourselves by insisting that what mathematicians do is by definition"not philosophy" and that "philosophy" is by definition what is done by people whose paycheck comes from a philosophy department of an academic institution.
If we accept this point of view, then we should have no qualms about using the word "philosophy" to describe the sorts of activities that you described as being "carried out exclusively by mathematicians and scientists." In that sense, philosophy has always clarified mathematics and will continue to clarify mathematics.
Now, you might still have a lingering question, which might be rephrased as follows: "Does the kind of philosophy of mathematics that happens to currently be the bread and butter of those people who happen to be paid by academic philosophy departments stand to offer any clarification of mathematics?" Phrased this way, we can see that it is more of a sociological question (and a historical question, since the political agreements about who owns what turf change over time) rather than a question about philosophy and mathematics per se. But still, we might be interested in the answer.
Generally speaking, the kind of thing that is covered in "textbooks of philosophy of mathematics" interacts with mathematics via the foundations of mathematics in general. The influence that this kind of philosophy has had in clarifying mathematics in general is clearest if we look back at the early 20th century. For example, today we enjoy an immensely clearer notion of what a "proof" is than anyone had in the 19th century, and that is thanks to the clarifying work of those who worked in foundations. We also have a much clearer notion of the distinction between constructive and non-constructive mathematics, thanks to foundational work surrounding the axiom of choice, intuitionism, uncomputability, etc. For a modern example, Voevodsky and his collaborators are now pushing homotopy type theory as their preferred approach to foundations. (Admittedly most of their paychecks don't come from philosophy departments but until recently, I think most authors of books and papers on type theory were paid by philosophy departments.)
In my opinion there is a continuum between foundations of mathematics in general and foundations of specific areas of mathematics (e.g., the foundations of algebraic geometry—think of topoi), but if you insist on defining "philosophy" according to which pot of money the paycheck comes from, then the foundations of algebraic geometry won't count because our society is set up so that such people experience pressure to migrate to the mathematics department. Thus so-called "philosophy of mathematics" is more-or-less forced to limit itself to clarifying the foundations of mathematics in general.