There might be a better solution, I just added a valid nameserver IP address to the ethernet gadget, and that worked.
As you highlighted, using internet nameserver cannot resolve name of wireless repeater and other computers of internal network. The Pi the ethernet gadget is plugged in uses the wireless repeater as DNS server, perhaps the ethernet gadget can do so as well? Alternatively perhaps the Pi where the gadget is plugged into can forward the gadget DNS requests ...
When ethernet gadget is plugged into a Pi that is connected to wireless repeater, its nameserver is set to localhost (/etc/resolve.conf), and name resolution on the gadget does not work: https://www.raspberrypi.org/forums/viewtopic.php?f=91&t=306121&p=1912854#p1912787
There might be a better solution, I just added a valid nameserver IP address to the ethernet gadget, and that worked.
As you highlighted, using internet nameserver cannot resolve name of wireless repeater and other computers of internal network. The Pi the ethernet gadget is plugged in uses the wireless repeater as DNS server, perhaps the ethernet gadget can do so as well? Alternatively perhaps the Pi where the gadget is plugged into can forward the gadget DNS requests ...