MagicDNS + resolv.conf + network disconnect => Broken DNS

### What is the issue?

So I have this Debian 12 box with 1.80.3 and this morning I noticed that DNS seemed broken.

After some investigation I noticed that /etc/resolv.conf looked like this:
```
nameserver 100.100.100.100
search my-chosen-name-redacted.ts.net

```

Tailscale was running but not logged in.

Then digging in journalctl I found this:

```
Mar 20 16:56:49 dahlia tailscaled[806]: health(warnable=no-derp-connection): error: Tailscale could not connect to the 'Frankfurt' relay server. Your Internet connection might be down, or the server might be temporarily unavailable.
Mar 20 16:57:01 dahlia tailscaled[806]: logtail: dial "log.tailscale.com:443" failed: dial tcp: lookup log.tailscale.com on 100.100.100.100:53: read udp 49.13.201.138:56709->100.100.100.100:53: i/o timeout (in 20.004s), trying bootstrap...
Mar 20 16:57:11 dahlia tailscaled[806]: trying bootstrapDNS("derp9b.tailscale.com", "144.202.67.195") for "log.tailscale.com" ...
Mar 20 16:57:11 dahlia tailscaled[806]: trying bootstrapDNS("derp24d.tailscale.com", "2001:19f0:c000:c564:5400:4ff:fe26:2ba8") for "log.tailscale.com" ...
```
and then continued with the derp lookups for hours.

A temporary fix was to manually restore my hoster's nameservers, but I might be missing something here. It could be that I did an apt-get update and reboot before it broke and then didn't notice because inbound worked fine.

### Steps to reproduce

No idea, somehow the connection got lost and the machine was stuck with just the 100.100.100.100 as a resolver, but that didn't resolve anything, not logged in. I would usually assume it won't break DNS resolution for the whole system, but fall back to the previous entries.

### Are there any recent changes that introduced the issue?

_No response_

### OS

Linux

### OS version

Debian 12

### Tailscale version

1.80.3

### Other software

_No response_

### Bug report

BUG-c4effeb06ce9979d6d7bc614fab53fd119522da1e87d800788887c1b5f4c5f12-20250321234335Z-37c2f6ccf9bc396d