My router can't do it

Logged in, the setup assistant fills in your own host name and token.

Many routers from internet providers, for example Telekom Speedport and Vodafone Station, only offer a fixed list of DynDNS services and no field for your own update URL. That is no problem: any device in your home network that is always on can do the job.

Raspberry Pi, Linux server or NAS: use ddclient or inadyn.
Anything with cron and curl: one line is enough, it runs every five minutes:

*/5 * * * * curl -fsS -u "<your-host>.dynr.net:<your-token>" "https://dynr.net/nic/update?hostname=<your-host>.dynr.net" > /dev/null

dynr.net uses the address the request comes from, which is your router's public address.

Port forwarding is still set up in the router itself: dynr.net only gives your address a name.