I'd argue that if the president is intended to be a position of power, then a separate election is good - otherwise, one party (presumably the largest) in the r

I'd argue that if the president is intended to be a position of power, then a separate election is good - otherwise, one party (presumably the largest) in the ruling coalition has power far beyond what their vote count would justify, and the person gets power without real input from the electorate. Especially with many parties, if parties A and B are reasonably similar, it's quite plausible to like party A's platform while preferring party B's presidential candidate. A separate presidential election accounts for that, allowing you to elect A for the house, and B for president. Note that parties A and B are similar - I'd argue that they'd have to be, and that it's extraordinarily unlikely to get a president the ruling coalition is incapable of compromising with. This works out because overall, the vote for president should be ideologically similar to the votes for congress/parliament. (That working, of course, is contingent on some form of ranked voting for the president, rather than the FPTP system the US has now.) All that said, I'm from Canada, and I don't really agree with the idea of a single person with large amounts of power, but that's my take if there must be one.