Ultimately we don't know much about that era of time, but I suspect it was less like fumbling around for millions of years looking for a light switch, and more like the gradual warming of the planet with warmer and cooler seasons/years.
Iirc at least one of the other things related to development of eukaryotes was that atmospheric oxygen had to first be generated by early cyanobacteria.
So maybe that proverbial light switch was being flipped millions of times through random encounters but only became more viable after the voltage (atmospheric oxygen levels) became high enough. Maybe that's the reason it took hundreds of millions of years, because transforming by bacteria just takes that long.
We just don't know unfortunately. However, we DO know about species getting wiped out by asteroids or human cultures getting wiped out by disease or conflict with superior cultures. Any of these filters seems more of a hurdle to me than the development of eukaryotes.