I think I understand why Avahi wouldn't work with that explanation, I keep digging and now my router is not exactly inaccessible, it's managed by my ISP instead which is annoying but I supposedly can still ask them to tweak things for me so I may have a chance
Tailscale already has DNS servers working on Tailnets (they call it mDNS, or magicDNS) which is I believe 100.100.100.100... actually, the documentation states that every tailscale-running device is a DNS server on itself too but either way that IP is private on every tailnet, remains accessible and host names are configurable on the Tailnet too
If I'm not mistaken on the functioning of DNS, I should be able to do it with that primary-router secondary-tailscale DNS setup I hypothesized and on LAN that will yield perfect connectivity, while out from home there would be no such records and it'll fall back to Tailscale's DNS which is already private (worst concern is just sending a request for a hostname on public network but that shouldn't be that big of a deal)
At this point I might have this solved, but of course, more input is nice too, It'll take some time for my ISP to work for me on this
Right, I didn't pick up on that m/multicast DNS difference, I thought I was talking about the same thing, thanks for clarification
Sounds like I do have options here, so while I wait for my ISP to comply, I'll also try out Avahi then