As one of those 'skeptical people', I find the above claim very hard to believe. Everything I've been able to find online regarding new nuclear designs seems to fall into one of two categories. Either it has a) serious security concerns (such as thorium reactors producing weapons-grade fissiles) or b) are untested designs that still have serious logistical or engineering hurdles to overcome (such as molten salt reactors, where they are still trying to engineer internals that won't corrode within a couple of years from the salt).
If there are 'totally automated and safe' designs out there that have been tested and are viable, I haven't heard about them. And considering the number of investors that would be climbing over each other to fund said 'magic bullet' energy solution the minute someone announced (with proof) that they had a 100% perfectly safe nuclear reactor design that did NOT produce insanely dangerous waste that needed to be stored for hundreds-to-thousands of years ... well, I think people are right to be skeptical about nuclear power. And for that matter, perfectly reasonable not to want current reactor designs (or their associated waste) anywhere near where they live or work.
You are correct that there are lots of walk-away-safe designs that for now are purely hypothetical, i.e. haven't been built. The reason investors *aren't* climbing all over each other to build them is because the regulatory, political (NIMBY!), and legal hurdles (note the conspicuous absence of "engineering" in that list) have an enormously negative impact on the ROI of a nuclear power plant. Yes, engineering is more demanding than a fossil fuel reactor, but it's not as large a contributor to delays as the other three. If you're interested in diving into it,
this video does a great job of explaining it in back-of-the-napkin numbers.
But the waste disposal issue is major. We are having issues with our present reactors and waste disposal.
Waste disposal isn't as large an issue as you might think. The spent fuel is almost always stored on the same site as the reactor, and actually ends up being a pretty small quantity, relatively speaking. The reactors only need to be refueled every couple of years, so over the lifespan of a reactor, the fuel doesn't amount to a whole lot, volume-wise. IIRC, it's the low-level waste that can be a problem to store long-term--stuff like clothes, tools, equipment, etc that is mildly radioactive. And when I say "mildly," I really mean it. It's slightly elevated from background radiation, but realistically, it's not actually very dangerous.
That stuff piles up to a larger volume, and out of an abundance of caution, it gets the kid glove treatment, even though it's not really dangerous.