I finally tested on a fresh install... and it works. So I dug some more, comparing the two installations, and realized at some point is a JS error on my regular install, linked to the chat integration plugin. In the end the code was outdated (it relied on a deprecated API), so I pulled the last version from git and... it worked.

Long story short: this was a plugin bug 🤷 Sorry for the noise, and thanks for suggesting testing a separate install!