For testing purposes, I installed No Script on my browser. This completely disabled the new editor box in a fashion that mimics what was described here. Whitelisting the Org solved the problem.

Regardless, this is clearly a specific issue with how individual users' browsers are configured and how they are interacting with the new editor. I'm not really sure what else can be done about this from the Org's side. If you can't fix this by upgrading/reinstalling your browser or by disabling whatever noscript-style addons you've got running, my recommendation would be to change over to the Basic Editor in the General Settings Message Editor Interface option.