The key difference is that in (most) European electoral systems, voters vote for MPs, who then in turn vote among themselves to decide on a leader for the count

The key difference is that in (most) European electoral systems, voters vote for MPs, who then in turn vote among themselves to decide on a leader for the country. This creates a (limited) incentive for cooperation when no single party holds a majority, since a coalition government is required. Simultaneously, voting for an MP has multiple impacts. Even if the MP isn't part of the coalition, they still have a say in how the government legislates, they can participate in debates, write bills, and take advantage of the resources of a government office to help their constituents. Finally, voting for an MP is a local affair. The barrier to entry at this scale is high, but tractable. Alternative parties can offer credible competitions at the town or county level (e.g., the US congress has had at least some "independent" representation as far back as I can remember). These three factors create a substantial difference in incentives in the US presidential election. There are no shades of victory: Even if your elector votes for a 3rd party, there is no long-term benefit. After they cast their vote, they're done. This makes voting for a 3rd party extremely high risk: you're gambling that everyone else feels the same way you do. Even if you don't like either of the main two candidates, you're faced with what amounts to a prisoner's dilemma where the other 300 million players have repeatedly shown a policy of 'always betray'.