I don't like the idea of this, but even the implementation is bad. If we're going to do DNS over HTTPS, then there should be a standalone application, and the system should be reconfigured to use it, so all running applications on the system use it.
I mean, do we really want all of our desktop applications to have their own built in custom ways of mapping domain names to IP addresses?
[edit] E.g on Linux, it could install an application with a DNS interface listening on localhost port 53, which would then convert the request into a "DNS over HTTPS" request, and resolv.conf would be updated to use that resolver.
I'm using Cloudflare's cloudflared [0] on all of my machines and it working well and does what you are looking for. Nice bonus is being able to collect metrics from each of the agents in Prometheus.