Personal theory of my own, that I have had for some time now. But you know, the United States of America was founded by former British aristocrats, land owners and merchants. I am not saying that is good or bad. It just happens to be so. So is it possible the American presidency is in many ways just based on the British monarchy? The US president has an unusual amount of powers. He is commander-in-chief. Heck, he doesn't even need the consent of Congress to start a war, really (cf. Vietnam). And the veto power. What other chief executive of really any other country has veto power, the power to basically nullify any law he doesn't like, at will? It seems clear to me I may be on to something. But should he have so much power? Think about it.
Technically the scandinavian monarchs, but I do say technicallly because the monarch has to remain politily neutral. And Trump isn't the one in power, no president or monarch is the people are they are the ones who vote, and can rise against a government/monarch...but I get what you mean
Ultimately what you're asking is; is there a better version of the USA presidency? You mention the US president has an unusual amount of powers. But why shouldn't he/she? The EU has that over the UK, just like the UK has that over Scotland/Wales/N.Ireland, just like England that has over every region of England, and so on. The current system has been means tested from thousands of years of human civilisation building by countless generations over blood and death, whilst our generation bicker over which political party had the overall majority of an election even though they did or did not win. I used to think the Queen was useless and totally irrelevant for modern times. But actually, all things considered, the monarch is a brutal task, as with any leader over a massive populace of people. These days success is not measured by success, but by failures. That is our problem. When shit hits the fan we'll be back to square one as a species, and this time we've got mother nature to combat and not other people.