IP > DHCP Server > Networks > DNS Servers

This is where you need to put your DNS servers so they're given to the clients (instead of your Mikrotik being used between your clients and your DNS servers).

This problem is presented in a confusing way. Pro tip: Spending the time to ask a well-organized question with all of the pertinent details sometimes gives you the answer before you ask. That said...

Okay, let's see if I have this right: You have a local network with a Mikrotik router as the gateway to the Internet, two Adguard DNS caching servers (one on the local network and one on the Internet), and a Proxmox server on the local network. The Adguard servers are pointed to 1.1.1.1 and 8.8.8.8, the DNS caching server in the Mikrotik router is pointed to the Adguard servers, and the Proxmox server is pointed to... what?

It sounds like—and this is just a guess—that the Proxmox server is getting its network configuration from the router via DHCP, which then points it at the router's internal DNS caching server. DNS queries go: Proxmox -> Mikrotik -> Adguard -> Google. When you turn off the router's DNS server (by unchecking "Allow remote requests"), the Proxmox server can't resolve any DNS queries for obvious reasons. (The DNS server it uses has been turned off!)

Everything works fine with the router's DNS caching server turned on, but if for some reason you want to turn it off, you need to point your local network devices (i.e. the Proxmox server) at a different DNS server. You usually do that by changing the DHCP settings on the router, so it sends DHCP clients the addresses of the Adguard DNS servers.