The whole term "base load" was invented to justify inflexible nuclear power plants. It is not special and the trifecta of wind, sun and batteries can provide it just fine.
Exactly. "Base Load" is a fallacy. Nuclear suffers the exact same problem as wind and solar. In fact, the industry has a term for it - it's called "non-dispatchable".
The term means a power source whose output cannot be synced to a load. If you turn on a light, the power plant needs to supply extra power for that light. And wind, solar and nuclear cannot do that.
Wind and solar are obvious why, but nuclear is also problematic because it takes hours to change its output.
Dispatchable sources include fossil fuel (natural gas especially), hydro, heliostat solar, as well as storage technologies (grid battery). These can vary their output in minutes which as far as the grid is concerned, sufficiently fast. Battery is especially fast and can react in under a second which can fix sudden grid lopsidedness.
Nuclear advocates want it, and it makes sense. But it's just another source of power that cannot be controlled. Grid batteries make nuclear more practical, the same way they make solar and wind more practical. Because it's actually a disaster if grid demand drops below what a nuclear plant is producing - you cannot shut down a nuclear plant and a grid with too much production will also shut down to protect itself.
Of course, the other problem with nuclear is heatwaves, as we've seen Europe struggle with production through their heatwaves when cooling water that is too warm to use, jellyfish, and other things. And remember, European nuclear power is far more modern than US nuclear so it's not old designs or anything.
So the US can have more nuclear power, that's no question as its deployment is far lower than Europe, but it's no panacea because the US can suffer with nuclear power the same way Europe does.