1) Everything in the universe, including the sun's solar radiation, is finite. It's just a matter of scale. The heat in the earth's core is practically infinite for humanity's current and near-term needs.
2) The heat generated in the earth's core is a combination of fission decay and residual heat from the earth's initial formation. The fission reaction is locked away 5000 KM deep (as in 5 million meters), deep-enough that it poses no threat even if you drill down thousands of meters to tap into that. That is very much not true of man-made nuclear fission.
3) There is still no long term storage mechanism that is guaranteed to keep nuclear waste for the several million years it would take to decay to safe levels, considering possible things that could happen over that time scale.
Wow, quite the oversell there! Modern nuclear reactors are widely recognized as safe by nonpartisan sources. And why do we need to solve for millions of years of storage? In a couple centuries at most, we'll be able to blast the stuff into space if that's what makes us comfortable.
2) The "residual heat" is long gone, and my whole point here was that saying nuclear was limited when geothermal is a form of nuclear was disingenuous/misleading. Trying to turn that into a scare / safety issue is exactly the sort of nonsense I was objecting to.
3) There are lots of viable long-term storage options, including grinding it up, mixing it with the tailings and putting it back into the same hole we dug it out of. Or dropping it into a subduction zone, and letting it fuel future geothermal plants. But we won't do any of those, because it's far too valuable. If you are worried about things being "misleading and exaggerated" you probably shouldn't be bringing up the nuclear waste "problem".