@drinkingbird thank you for confirming this behavior of the Asus FW/DHCP server.
This is a stupid design decision from Asus (it's 2023, and this function doesn't eat much CPU or memory). What's even more annoying is that it is not stated "if you manually put in a DNS Server, then NO SECOND DNS Server info is sent to DHCP clients".
As I said, another "design flaw" from Asus in a growing list of design flaws of Asus in their routers. No VLAN support is another one, not to mention the often-reported WiFi problems (some clients just can't connect, no matter what you do) or the privacy-invasive TrendMicro stuff (at least that's optional).
After years of wrestling with Asus to get what it should do, and buying several routers from them, this is it for me, the nail in the coffin. I'll only use Asus for secondary or tertiary stuff, but no longer as my primary router.
EAP670 got my WiFi issues fixed, and OpenWRT-based stuff (or OPNsense) will to the routing/firewall. So sad, Asus :-(