[krick]

> people are voting for whoever they think is their best state government

It's even more complicated than that. For instance, "head of state" of Latvia in the Council which nominated current EU Commission president in 2019 was Karinsh, the prime minister of Latvia. Not being directly elected, the prime minister of Latvia gets nominated by president (not directly elected as well, BTW) and approved by coalition in the parliament. The punchline is that he was the leader of a party that got the least number of votes (and hence, number of seats in the parliament — 8/100) in the corresponding election. The party that got the most votes (and most seats) was not included into the coalition at all.