I am not sure I understand your question.
Many modems also act as routers, so they do have a built-in DHCP server and give out IP addresses in the private ranges.
If you connect a router to a modem without using the WAN port on the router, you are essentially using it as an access point/switch (not as a NAT router), hence its DHCP server and NAT routing functions don't play a role.
Keep in mind that every time you connect a new device/MAC address to the modem you will have to reboot it, so that it recognizes the new device. Why not reboot the modem?
The "DHCP Release/Renew" function in the router will only expire the IP leases of dynamic DHCP clients connected to it directly, it will not affect the connection between the router and the modem.
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