I myself do two types of statistics.
One is statistical analyses of psychological experiments. I agree that the classical NHST paradigm is... problematic. The (almost invarably misunderstood) p value gives the probability of observing a test statistic as extreme as the one actually observed under the null hypothesis, conditional on the model being correct. Now, we are pretty sure that the null hypothesis is not true, otherwise we would not be running the experiment. And the model is almost certainly not correct, given tapering effect sizes in the sense of Burnham & Anderson. So NHST can be said to give an answer to an irrelevant question using an incorrect approach. Yes, I understand how this makes mathematicians queasy.
Then again, it works reasonably well, even in light of the so-called replication crisis. Once you look beyond single studies, a number of things simply fit together too well and make for a coherent whole. So the statistics seem to be doing something right. Also, as a mathematician, one understands the limitations of incorrect models, and the power of low-order approximations. The Taylor rule could be said to hold.
My day job is forecasting retail sales. I mainly use regression type approaches with a dash of Bayes, nothing one would reasonably classify as ML. No p values involved. And here, I trust my statistics because they work. The forecasts are not perfect, but they are again reasonably good. At least good enough to ensure that enough product is on the shelves when the next promotion and price reduction starts.
I studied pure math, and as I said, I understand the misgivings of your professor. Statistics, along with applied math and physics, have an interesting in-between existence. On the one hand, you can derive theoretical results, some of which actually help in the "real world". There is really a bit of art to understanding how far the theory is applicable and when it stops being so, and where to leave pure theory behind in order to achieve results that matter outside it.