The most direct problem is styling issues. Cross-component CSS in either direction has some serious limitations. I've written a little bit about it[1] in my blog but the short version is that there are some things that simply become absolutely impossible when using web components.
My main other gripe with them is the need for a build phase. The nature of WC almost begs for them to eventually become zero-build-time, but right now this just isn't practical. It requires too much boilerplate in every .html file (utf8 is broken on my site), the syntax isn't natively there even with tagged template literals, and there's no concept of data-list-fetching or data-based file generation.
There's a branch on my personal website[2] where I tried to start using web components, and it was so problematic that I long abandoned it.
Overall, I abandoned web components entirely in favor of making my own customized JSX-based SSG from scratch[3] which solves the same problems Lit, Next.js, et al. are intended to solve, but in a completely different way: using components for convenience, conciseness, and reusability, but only at the build-phase time. So far it's a well kept secret, which is probably good since it's been evolving so quickly that I wouldn't have been happy with any iteration being widely adopted so far. (Though I think this morning I finished off most of what I was unhappy with.)
[1]: https://sdegutis.github.io/articles/2023-08-07-modern-90s-we...
[2]: https://github.com/sdegutis/immaculatalibrary.com/tree/reset...