That's not at all how the EU works

Because they don't rule anything in the individual countries. The EU is subject to treaties among the members, currently the Treaty of Lisbon, ratified in 2009. They typically need to have unanimous consent to do anything with the force of law.

Since they're in the news, as just one example I invite you to compare the current governments of Sweden and Hungary.

There is absolutely no comparison to the President of the United States in any capacity in the EU. The President of the European Commission is a completely different position with a completely different purpose. It's an appointed position. The EU also has no military aspect whatsoever (it can provide military aid, as it has to Ukraine, but it has no military). And there is still no ratified European Constitution. If there ever is one (the last vote on one failed in 2006 when France and the Netherlands voted it down; there were other member states who hadn't voted yet), it's likely a very long way off.