I am using i3 but what bugs me is that tabs now seem to be part of the browsers.

There's no elegant way of telling Firefox to never use tabs, always open a new window. There are some extensions pretending to this, but the experience is terrible. There actually is a new tab created which is then detached.

I don't usually use Chrome, so I looked up wether it handled this, but it seems to rely on extensions too.

Qutebrowser works, but it has other limitations, such as poor extension support.

I've had the exact same experience you described. Realizing i3 is the way to go and apps shouldn't manage tabs themselves - trying to stop Firefox / Chrome from doing so - realizing it's not simple and the no tabs extension is problematic. Trying qutebrowser and giving up because it's limited.

I've settled on NoTabs Firefox with a userChrome.css that hides the tab bar until Firefox someday supports native no tabs.