New nuclear power plants and current reprocessing can absolutely make this a non-problem. There’s plenty of radioactive material.

That doesn't matter. Uranium isn't that rare, and we only need a power source that kicks the can down the road long enough to build renewable alternatives. Once we start running out of uranium, its price will increase and it'll be profitable to replace nuclear plants.

The only reason the same logic doesn't apply to fossil fuels is that climate change is an externality that will kill us long before we run out of fossil fuels.

If nuclear solves the problem for a century, then in 2126 our solar, wind, batteries, enhanced geothermal, wave etc will be very mature and able to replace nuclear quite easily.