Base load is a concept of the past, grids around the world are being redesigned to be flexible to reap zero-production-costs renewable energy. Nuclear (which is impossible to run economically as a flexible asset) simply does not fit into that new world anymore.
It'd be way easier to build a few nuclear plants than it would be to build an equivalent constant energy source from solar+wind and batteries. The nuclear plants would also consume far less land area.
True. I'm biased by living in a place where the peak load does happen during daylight hours (because that's when you need to run the A/C) and where heating usually happens via gas. Electric heating would indeed shift that dynamic (though municipal water heating would shift it the other way).