I recently tried to enable system-wide DNS over https on Fedora. To do so I had to to some research and found out how comfusing it is for the average user (and even experienced users) to change the settings. In fact there are multiple backends messing with system DNS at the same time.
Most major Linux distributions use systemd-resolved for DNS but there is no utility for changing its configuration.
The average user would still try to change DNS settings by editing /etc/relov.conf (which is overwritten and will not survive reboots) or changing settings in Network Manager.
Based on documentation of systemd-resolved, the standard way of adding custom DNS servers is putting so-called 'drop-in' files in /etc/systemd/resolved.conf.d directory, especially when you want to use DNS-over-TLS or DNS-over-https.
Modern browsers use their buit-in DNS settings which adds to the confusion.
I think this is one area that Linux needs more work and more standardization.
How do you think it should be fixed?
You said all browsers would follow your system DNS, I just explained that's not always the case.
And there is actually a common problem with devices on the LAN that use DoH. You can block their access to the specific DNS servers they use, or block their access to the internet altogether, but you can't force them to use your DNS settings.
Both Firefox & Chrome follow my system DNS at default settings. Just because Firefox forcefully enrolled US users to Cloudflare's DOH doesn't mean that DNS is broken for every one else.
Again. Has nothing to do with the topic i.e Linux DNS. Applications can use their own custom DOH/DOQ resolvers to bypass system DNS, this has no bearing on the brokeness or not of systemd-resolved or any other system DNS resolver.