I'm not really familiar with troubleshooting DNS connection issues, but it sounds to me as though it would be beneficial as a first step to try different DNS providers in order to determine if there might be an issue with the provider, and not the router or ISP.
Dan-cer The time required to optimize everything and get it up and running was considerable: Added up to about 2 weeks of more or less intensive work.
This sounds rather exhausting. Perhaps it would be less painful to take your privacy "journey" a bit more gradually? It's what I prefer when starting on new projects in my spare time. For instance, I've started writing a novella, which I work on whenever I want to (and when time allows). But if I get exhausted from writing or thinking about it, I find it better to just take a break. It's not a privacy thing, but it's still a chosen project.
Dan-cer One week of this was dedicated directly to the GOS,
I'm wondering what you spent time on that took a whole week of your time just to set up GrapheneOS.
From personal experience, not much is needed to get GrapheneOS "up and running": installing the OS, getting your favorite apps, etc. The OS has sane security and privacy defaults which work without causing usability issues for most people.
It sounds as though you want to go considerably further, given that you find it intensive. I haven't seen Brockwell's videos, but they sound more like tips on what to do given certain threat models, rather than something necessary to achieve some goal of "privacy".
Dan-cer Nevertheless, I imagine that it would be helpful if a proven private DNS is already implemented in the GOS
My thoughts on this: there are several points to consider if an OS with high privacy and security standards is to include a default or a selectable DNS within the OS. Such as:
Speed and latency: users are located throughout the world, and a provider's DNS servers might have high latency in some areas and with some ISPs. Consequently, the user experience will vary, and feedback on latency and speed will ideally need to be collected from the user base in order to select a provider that works well for most users.
Privacy: which privacy criteria should the OS developers consider? I'm assuming that most users would prefer a DNS service that does not keep connection logs. How would the OS developers verify this? Would an infrastructure and/or server configuration audit by an independent party suffice? And if so, how regularly should such audits be completed in order to be reasonably sure that the DNS service does not start keeping connection logs? And what if it is revealed that the service out of the blue has started logging connection data despite their promises? How would the OS developers then feel towards their user base? They would have to ship an OS update to either revert to the standard DNS service of users' ISPs, or ship an update to replace them with new DNS servers, which would have to be chosen rapidly.
Uptime: a DNS service would need a high level of uptime – which is especially important if it is set as the default DNS service of the OS. If not, users who are unaware of how to troubleshoot internet unavailability would understandably become annoyed. The OS could include fallback DNS servers, but then those would have to be vetted against the chosen criteria as well, adding further work.
My personal opinion is that it's not productive to ship custom DNS servers in the OS. Even if they are not the default and only selectable within the Private DNS selection dialog, I imagine several users would consider that a recommendation of those services on behalf of the OS. I certainly would.
In any case, the project already recommends using the network-provided DNS servers if the purpose is to "blend in with other users":
Using the network-provided DNS servers is the best way to blend in with other users. Network and web sites can fingerprint and track users based on a non-default DNS configuration. Our recommendation for general purpose usage is to use the network-provided DNS servers.
Please note that I am not associated with the GrapheneOS project.