The domain verification is an essential step to ensure that the added domain is valid, not expired and you have sufficient privileges to manage the email service of the domain. It also proves that you have access to the domain or the domain’s DNS manage and have the information about the users of the domain and other significant information required to change the email hosting.
There are several way to verify your domain. But in case you have registered the domain with a provider and changed the domain name servers to a different provider, then the DNS records should be changed in the other provider. Basically when DNS records are looked up from your domain’s name servers, you may get clarified with your registrar of where your domain’s name servers are pointed to.
Once you have verified the domain, the CNAME or the HTML file can be removed and is no longer necessary. Once your domain is in verified state, in case you switch your DNS provider you need to check the MX records of the domain alone. Most DNS providers reset the MX entries during DNS shift or change.
If you are a multi-tenant user and manage sending domains for your own users, you’ll need to work directly with your users to verify their sending domains. E.g. to verify a domain you need to have login access to your email account where the sending domain is stored.