"The problem is simple: third party candidates rarely win any electoral votes, but they do get plenty of votes from the electorate."

The plurality voting system promotes the eventual establishment of two major parties and noncompetitive minor parties. Replacing plurality voting with ranked choice voting would allow third parties to grow and be competitive.

https://www.youtube.com/watch?v=s7tWHJfhiyo

"Second, supporting run-offs would prevent unpopular candidates from winning the election."

I disagree. Closed partisan primaries followed by run-offs usually result in the least undesirable candidate being elected instead of the most desirable.