Does ed25519 work on OpenWrt 19.07.7?

I want to upgrade to ssh-ed25519, but I see mention that it will not work on current OpenWrt version (19.07.7). At the same time the wiki says that it is necessary to rebuild the package to support it. Who to believe in this situation?

Both statements are probably true; I'm not sure where you think the contradiction is?

The crux is default OpenWrt images don't support it; but you can recompile and enable support in Dropbear yourself.

It turns out that there is no way to make do with this by rebuilding a separate package and you have to rebuild the entire image?

Sure there is. You just need to recompile dropbear and install that over the default package on your router. No need to build a completely new image. You still need the buildroot for that though.

All right! I'm looking at the pages right now

and I don't understand where to enable ed25519 support?

That's because you're skipping too much stuff. Buildroot documentation (to be read from the top):

This details how to configure the buildroot.