I am seeing this issue with a new installation of Remote Desktop 10.9.10 on macOS 14.6.1. The problem is not with the DNS server - it is with the Remote Desktop client querying the wrong DNS servers.
Using Wireshark, I can see that DNS queries are not being sent to the two DNS servers shown in macOS as learned via DHCP - instead, Remote Desktop (and only this product) is sending DNS queries to two DNS servers that we decommissioned last year, and which have not been learned via DHCP since then.
Why is the Remote Desktop client not using the macOS DNS server settings?