The ideal solution would be something that can be installed in any home router, but that's a non-starter because of total lack of home router configuration standards. Flashing a router is not user-friendly for anybody.
The next best solution would be your suggestion, but that too is not practical unless it comes preinstalled on all OSes because the average user won't even know they need something called a DNS proxy. I don't see OS vendors agreeing to preinstall it. Additionally, AFAIK, popular OSes like Android and iOS prior to some versions don't even allow such system-wide DNS proxy configurations.
The practical approach left then is to implement it in browsers, solving it at least for the most common use case on all devices. Everybody knows how to download and install and use browsers. In a discussion forum I frequent, average users ask for ISP censorship bypass solutions all the time. Since Chrome does not support DoH yet, among all the possible solutions - VPN, Tor, SSHproxy - using another browser is actually the most user-friendly, least expensive, most performant option. It helps that it's Mozilla's product because their trust perception is higher than Google/MS/Apple/VPN providers.
I feel Mozilla's taken a good approach overall within the scope of their area of expertise.