The chicken and the egg problem is resolvable. A third option in a first-past-the-post voting system is not.

Maybe less likely, but it still happens. E.g. Weimar: very fragmented scenario (there were at least 4 major parties, not including the Nazi), still a lunatic won. And don't get me started on the Italian experience, or the French republics... In practice, the more fragmentation you have, the more likely that the electorate will eventually converge towards a strongman lunatic who will "clean up the mess/corruption/etc etc".

The safeguards against lunacy must go well beyond the number of parties, and still might not actually work.

It makes it easier to have lunatics win power without winning the popular vote.

Not sure that's true. If you have N parties and N/2 submit lunatic candidates, are we better off? I mean, I guess we have N/2 sane ones to choose from, which is an improvement, but we also have N/2 lunatics, any of which could win.