I am trying to have my domain that I purchased ("example.com") and I was following the steps off this youtube video . Where the person has to use the custom URL option after clicking the 'more/3 dots' option. Problem is, is that he has a completely different UI/Layout than me so I cannot follow his steps completely to publish my site with my domain name as its url. The following images shows his 'more' options then my 'more' options. You will notice he has the 'custom URLs' option and I do NOT have this options.
If you're using a GSuite account the custom URL is configured from the admin panel
I'm guessing that the problem is simply: that interface is not available if you don't verify domain ownership as your Google Sites site owner account. There can be more or fewer steps depending on the details of your situation, but here is an overview:
- Publish your site.
- Choose or add your property in Webmaster Central.
- Complete the subsequent steps to Verify it is yours.
- Perhaps leaving Google for the moment, in your domain host account, enter the desired subdomain like "www.yourdomain.com" as a new CNAME dns entry, with the value ghs.googlehosted.com to point it to your site.
- Return to your Google Sites site and check the "3 dot menu" again.
See also Google Help Center: Custom URL.