DoH isn't done by the OS. But that's my point. In order to use DoH, you have to (purposefully) use an extension/browser addon/browser setting.

As a system admin myself; if user applications started overriding the DHCP DNS that I give them, not only could intranet sites be broken, but I'd start having fights with users about it.

Edit: Rather, not overriding but querying the DoH instead of the provisioned DHCP DNS. I'm no expert in DoH, or how any of that works under the hood.

Further, when/if browsers turn on DoH by default, then I can't really fight users, because they did nothing wrong but use a browser. Suddenly, I can't support a browser or two because of it.

DNS caching by the application is fine, because they made the request to the OS, and got the response. That being said, TTL might be violated by that, since the record has a TTL, and whatever the application cache TTL is.