Some examples: text rendering is awful on some common browser configurations (mine included—Firefox on Linux/Wayland, rendered at 2× scaling and downsampled to 1.5×; the text is extraordinarily fuzzy, and it’s rendering in its own font because I suppose it’s bundling a complete font, shaper and renderer—since I force my own font selections so only the generic font-families work, and I also block all web fonts at fetch time); the font rendering technique means no fallback which means things like emoji would only render if they provided a font that supported that (at present they don’t, and it’s just a nasty technique all round, you want to be able to use local fonts); links are fake, so various interactions can never work correctly (e.g. Ctrl+click and middle-click should mostly open a background tab, and right-clicking should give you the normal context menu); scrolling can’t be done properly and will thus inevitably be extremely obnoxious on most devices; mobile-style pinch zoom cannot be supported.
⸻And I’m just mentioning some of the unfixable problems with this approach; there are more things that are wonky or painful apart from that: things that could be fixed, but where it’s not surprising that they’re broken because that’s a common problem when you throw away almost all of the stuff the platform gives you and try to implement it all from scratch.
I very strongly recommend against using this general approach at all on the web. The web platform just doesn’t provide the primitives required to make it anything other than miserable.
(I’m not speaking against egui for platforms other than web, only for web. The fact of the matter is that functionally you’re fine building a native app with native tech, a native app with web tech, or a web app with web tech, but that building a web app with native tech is a terrible idea that will always be inferior, in important, unfixable ways.)