It strongly depends on your threat model. For instance if you are concerned about privacy because you have a determined stalker, then your best option for email could very well be a new Gmail account, simply to blend into the "crowd"
As for me, the primary threats I seek to mitigate by using my own domain are:
Automatic profile building. If a company or service is using my data to build or add to a marketing profile, or sharing my information with other companies, they are almost certainly using my email address as the primary key to identify said profile. By using a unique email address for everything, I can ensure any such profiles will remain separate, hopefully only contain information which I choose to allow that organization to have, and I will know what organization shared my email if I start getting emails from an organization I did not initiate contact with
Data breaches. If a service or website I use is breached and my information stolen from them, then using a unique email address everywhere will somewhat segregate other accounts of mine against automatic attack based upon the breached data.
Also, using a catchall box on a custom domain does not preclude using other addresses as well. Often if I'll be doing something that has a different threat model, so I'll spin up a burner account dedicated to that purpose. But for daily life the benefits of my own domain far outweigh the disadvantages.
As for posteo's objections, most of those can be mitigated. You can register a domain fully anonymously with a service like njal.la There is also nothing stopping you from registering an account with a provider under false information, and/or use a privacy advantaged service that requires only minimal to no information. Or as a final option host it yourself.
You will always have to have MX records pointing to your provider, but without your own domain, just the domain on the address points to your provider without having to look up MX. So either way people will always know your email provider given your email address.
Again it all depends on your personal threat model and goals. If you are trying to be fully anonymous, then yeah a domain is not the easy way to go. If you are just trying to mitigate a lot of everyday privacy threats and have a reasonable level of anonymity, then it can offer a lot of advantages.