"The problems with today’s reactors, such as the risk of accidents, can be solved through innovation." - this is an example of wishful thinking. We don't know how we would solve the problems, we are just assuming someone will figure out a way. That assumption is probably influenced by survivorship bias. There's a bunch of technologies where the problems weren't solved, but we don't hear about them much.
The mean time to build a reactor is about 7.5 years[1], so if you wanted to transition all remaining fossil-fueled electricity generation in 12 years, currently with 120,000 TWh of fossil fuel consumption[2], assuming 1GW of generation per nuclear power plant, you would need to start building approximately 14 thousand (edit: originally read millions, corrected as pointed out in comments) nuclear power plants today.
[1] http://euanmearns.com/how-long-does-it-take-to-build-a-nucle... [2] https://ourworldindata.org/fossil-fuels