Do not host your own DNS nameserver if you don't know what you're doing. It can and will be abused into all kinds of DNS attacks.

See if your registrar offers a DDNS service. Alternatively if they offer an API you can update the record yourself with a script, or use a DDNS container app, or use a DDNS plugin for OpenWRT etc.

You can also separate the nameserver service from your registrar, like you've been doing with Hetzner, but you don't have to host it, there are ready-made DNS services.

One very good service which is free and has an API is deSEC.io. It's also been around for a while and is supported out of the box in most DDNS tools, and it's run by a German organization with a focus on privacy.

The catch with deSEC ID that they require you to enable DNSSEC, because their mission is similar to Let's Encrypt — to promote the use of secure DNS. It's not hard to enable DNSSEC, they generate and maintain all the records for you, naturally, but you'll have to enable it manually at the registrar, and remember to disable it temporarily during transfers.

Another good DNS service (with API) that comes down to $1/mo and also includes CDN services is bunny.net.