Forget about free domains if you want to do something remotely serious with them. A real domain is few dollars/year and AFAIK you could use it coupled with GitHub Pages for simple, static websites or with altervista.org if you need WordPress.

https://neocities.org/ might work for you.

I feel like there's an inherent misunderstanding in your query. If you want to run on your own domain - you run your own stuff (for the most part). You need a webserver, that's about it.

I dat to check google sites first since I never used it before.

From the quick glance I gathered, I‘d say there is no open source equivalent with free custom domains.

Registering top level domains is not available for free I think. The closest I know is duckdns.

If google is giving out top level domains for free now then it is a market dominance strategy because nobody else can do it I think.

Let me know if you have any more questions.

Hugo, Jekill, Docusaurus, host on github pages, netlify. Buy a domain for 15$ per year.

You want a website? https://wordpress.org/ is open source