Interesting Cloudflare Pro related problem
As far as I could test, it seems like Cloudflare Pro, among other cool things, prevents editing custom HTML WordPress widgets.
When CF is disabled (grey cloud, working only as a DNS, not proxy) - everything works fine.
Likewise, websites that don't use Cloudflare Pro (only the free plan) are not affected.
A bit more details - will keep it updated:
This could be a rare problem - I suppose not many people use a similar stack, so it may have gone unnoticed.
Mostly WordPress ™