That’s not how it played out in Buffalo NY in their last mayoral election. A socialist won the primary and the Democratic party launched a massive write-in campaign to keep their preferred liberal “Byron Brown” in power, to the point they were handing out stamps at voting locations so that they could be used to easily “write” him. They succeeded in keeping the socialist candidate out, and despite winning, their liberal candidate ended up resigning mid-term to take a cushy C-suite job instead.