Physicist here.

> From what I understand, the parts of quantum mechanics that aren't still fully understood are where it overlaps with relativity.

Yes and no. You're right, one big question are high-energy completions of the quantum field theories that make up the standard model and which we expect to break down at very high energies. On the one hand, this expectation is due to the fact that general relativity is likely going to play a role at these scales, so we'll probably have to solve the puzzle of quantum gravity along the way. On the other hand, though, there are many more reasons why the quantum field theories we have right now are expected not to be the full story:

1) For reasons of, well, beauty. Right now, the standard model has many free parameters but even if you ignore the parameters for a second, it's also a somewhat random collection of specific gauge theories (the latter is basically another word for quantum field theory), i.e. specific gauge groups. Therefore, people hope that there is an underlying "grand unifying theory" (GUT) that should become visible at high energies. Note that the beauty that comes with a GUT is only in parts what Hossenfelder means when she discusses "beauty".

2) More importantly than 1), the usual interpretation of the renormalization procedure in QFT these days is that it is a way to exclude physics at the very high energy scales from our calculations/predictions. Put differently, without introducing a renormalization scale (basically a "maximum energy") the equations we have blow up and people take this as a sign that our theories are incorrect at extremely high energies. So in this sense, current quantum field theories are nowadays seen as effective theories (which are only valid at low-energy scales) of an underlying "complete" theory which will hopefully cover all energy (and thus length) scales.

But apart from the lacking high-energy completion, there are more reasons to believe that we don't fully understand nature yet and that we might need to come up with a new theory:

3) Even after decades of research we're still lacking a rigorous mathematical underpinning for practically all relevant QFTs. Considering that nature so far has always been governed by laws that we could express in precise mathematical terms, this might suggest that we're doing something wrong at a very fundamental level. A lot of people (or at least those I have spoken to) hope that a theory of quantum gravity will solve this (and thereby reestablish mathematical sanity in high-energy physics).

4) Dark matter: As you might now, dark matter is a general name for the apparent stuff that causes our observations to differ slightly from the laws of gravity that we know (i.e. from general relativity). The existence of dark matter doesn't necessarily mean that we'll need an entirely new theory but if we're right in our assumption that dark matter consists of particle excitations of another quantum field (usually called WIMPs), we'll need to at least extend the current standard model.

5) The foundations of quantum mechanics: More specifically, the measurement problem together with the issue of having different interpretations of QM and, possibly, the issue of time. These are basically the issues danbruc meantioned above. For some background info see for instance Steven Weinberg's lecture on "What's the matter with quantum mechanics?" (discussed by Sabine Hossenfelder here: http://backreaction.blogspot.com/2016/11/steven-weinberg-doe...) and Carlo Rovelli's recent book (https://news.ycombinator.com/item?id=17376437). Again, some people hope that these issues, too, will eventually be solved by a theory of quantum gravity but this is anything but clear. (By the way, a friend of mine working on Bohmian Mechanics recently told me that there might be experiments to distinguish Bohm's interpretation from the Copenhagen one, so we might not even need to wait for quantum gravity.)

Anyway, let's assume for a second that we really need a full quantum theory of gravity to solve all these issues. Then, what's our best shot at coming up with a theory of quantum gravity? Well, studying the high-energy behavior of particles and hoping to discover "new physics". So in this sense, you were absolutely right of course. I just wanted to point out that it's not reconciling relativity and quantum mechanics alone that drives us.