Why do I need to add nameservers to resolv.conf?

I'm running Ubuntu 18.04 (upgraded from some earlier version) which uses Network Manager and systemd-resolved for name resolution. When I boot, my ethernet connection enp0s31f6 is brought up by Network Manager and given three nameserver addresses via DHCP, 10.1.13.10, 10.1.141.10, 10.1.13.36. Running nmcli shows the three nameservers under "DNS configuration". Running systemd-resolve --status shows them under a "Link 2 (enp0s31f6)" section. I can ping each one. No other connection is active.

testuser ☼ systemd-resolve --status
Global
          DNS Domain: (my org's domain)
          DNSSEC NTA: 10.in-addr.arpa
                      16.172.in-addr.arpa
                      168.192.in-addr.arpa
                      17.172.in-addr.arpa
                      18.172.in-addr.arpa
                      19.172.in-addr.arpa
                      20.172.in-addr.arpa
                      21.172.in-addr.arpa
                      22.172.in-addr.arpa
                      23.172.in-addr.arpa
                      24.172.in-addr.arpa
                      25.172.in-addr.arpa
                      26.172.in-addr.arpa
                      27.172.in-addr.arpa
                      28.172.in-addr.arpa
                      29.172.in-addr.arpa
                      30.172.in-addr.arpa
                      31.172.in-addr.arpa
                      corp
                      d.f.ip6.arpa
                      home
                      internal
                      intranet
                      lan
                      local
                      private
                      test

Link 3 (wlp4s0)
      Current Scopes: none
       LLMNR setting: yes
MulticastDNS setting: no
      DNSSEC setting: no
    DNSSEC supported: no

Link 2 (enp0s31f6)
      Current Scopes: DNS
       LLMNR setting: yes
MulticastDNS setting: no
      DNSSEC setting: no
    DNSSEC supported: no
         DNS Servers: 10.1.13.10
                      10.1.141.10
                      10.1.13.36
          DNS Domain: (my org's domain)

However, when I actually try to resolve a name, even the name of one of the nameservers, dig claims that "connection timed out: no servers could be reached".

testuser ☼ dig dcpdc001.(my org's domain)

; <<>> DiG 9.11.3-1ubuntu1.1-Ubuntu <<>> dcpdc001.(my org's domain)
;; global options: +cmd
;; connection timed out; no servers could be reached

Note that this name should resolve to 10.1.13.10, the first nameserver.

I have configured resolvconf to use dynamic updates. /etc/resolv.conf points to /run/resolvconf/resolv.conf. This file contains only (non-comments):

nameserver 127.0.0.53
search (my orgs local search domain)

If I add nameserver 10.1.13.10 to this file manually, suddenly dig can resolve again, and anything else that needs to see local names can do so. Removing the nameserver breaks that again.

I don't know much about the servers. They're part of a Windows-based network, but I can use them if I edit resolv.conf manually so I don't think that's the issue, and it implies I don't need to be authenticated to the domain to use them. (I can authenticate to the domain via Ubuntu using Realmd/SSSD, but not if I can't resolve the domain controller...)

The journalctl entries for systemd-resolved show only a few messages about "Using degraded feature set ... for DNS server" but they only refer to the third nameserver, not the others. Nothing for the primary nameserver.

How can I get name resolution working without having to manually edit resolv.conf every time I boot?

I assume the contents of my resolv.conf mean that Network Manager or Systemd has some sort of local caching resolver running? If so, would bypassing it fix things?

I increased the logging level of systemd-resolved and journalctl -f -u systemd-resolved shows:

Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Got DNS stub UDP query packet for id 19836
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Looking up RR for dcpdc001.(org domain) IN A.
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Switching to DNS server 10.1.13.10 for interface enp0s31f6.
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Cache miss for dcpdc001.(org domain) IN A
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Transaction 12728 for <dcpdc001.(org domain) IN A> scope dns on enp0s31f6/*.
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Using feature level UDP+EDNS0+DO+LARGE for transaction 12728.
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Using DNS server 10.1.13.10 for transaction 12728.
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Sending query packet with id 12728.
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Processing query...
Jul 20 10:33:23 heerij-ubuntu systemd-resolved[2352]: Timeout reached on transaction 12728.