This reads entirely as "everyone else's approach to problems have been unsuccessful, therefore they should use my approach instead, which also had been unsuccessful solving the same problems"
That's not entirely fair. I've been reading her blog for a long time.
She's basically opposed to maths in physics for the sake of beautiful maths (nature doesn't have to be beautiful in our eyes), and spending vast amounts on money on bigger particle accelerators when there is no particular theory to test.
She wants to see real progress in her field. This doesn't necessarily make her right, but she has valid viewpoints that somewhat threaten the status quo of particle physics.
That may well be the worse example you could have given, seeing as how special relativity was necessary to reconcile Maxwell's equations with Newtonian mechanics, a major inconsistency in physics.
Perhaps a better example could have been GR explaining why gravitational mass and inertial mass happen to be identical (except that GR also solves other real problems - notably, how to apply special relativity in the presence of gravity).
If you read the literature of the time, the issue was considered as largely understood. I would highly recommended science and hypothesis by Poincare (1905). You'll see a lot of concepts and equations that you might have assumed came from special relativity or even general relativity.
I will be interested in reading that, but my understanding in general is that the previous (physical) theories were a mishmash of hypotheses that were patching up the luminiferous aether model as new experiments were invalidating the older assumptions. In contrast, Einstein's paper explained all of the experiments in a single simple theory with a minimal set of extra assumptions.
Additionally, we already know that the luminiferous aether model would have soon hit other major problems with quantum mechanics, which was already starting to be intuited by Einstein and others. So even if SR would have been seen as just a mathematical curiosity at the time of publishing (solving a non-problem), it would have soon become even more important.
Very importantly, SR also had much fewer assumptions (free parameters) than any of the other successful models it replaced. Contrast this with String Theory, which has numerous free parameters, doesn't fully explain currently-known phenomena, and has accumulated more than thirty years of study from a vast amount of physicists. SR was one physicist's individual work for a handful of years, and the moment it was presented, it was a short and complete theory that could immediately replace more complex ones.
If any individual physicist wants to similarly individually investigate some of the non-problems listed in the article, and were to come up with a new formulation of QM/QFT that explains them with a similar theory - then by all means, I am certain Dr Hossenfelder has no problem with that. It is when the search for such a theory becomes a decades-long approach sucking up most of the funding in the field with no results at all that it becomes a problem.
The equations that students must often learn for special relativity were already known, hence the Lorentz naming. The question at the time was largely about how this undetectable substance called the ether behaved.
Another example might be the "discovery" of anti particles or of black holes. Both of which were long regarded as just mathematical curiosities.
Also I'll add that there were many "outs" for those that supported aether theories such as frame dragging or simply experimental error as those themselves doing the experiments often believed and kept trying to develop more refined experiments to prove the existence of the aether even into the 20s
