When you use a modern browser, your lightweight distro doesn't matter anymore. The web stack is heavy, even if sites are coded with performance in mind.
I repurposed an old laptop for occasional video watching, running a small debian. Firefox or chrome are huge compared to everything else.
The other factor is that high video resolution eats the machine.
youtube-dl (480px) + mpv is much nicer than streaming in a browser.
Most websites still work OK on slow machines (my travel laptop was one and it could handle 90% of the websites just fine). OS/Desktop env is often the biggest sink. That machine for example would hit 75% CPU on windows 10 just idling which didn't leave enough room for anything else. Wiped it with Ubuntu + Gnome and it went to ~25% CPU usage while idling. Switched to Ubuntu + XFCE less that 3% while idling.
The only website I had a problem with was Patreon.