Transitioning Email is a hard thing to do. I suggest doing so gradually in a multi-step process:

• Start using disposable emails for signing up for things TODAY. Addy.io, duckduckgo, mozilla, proton, whatever you trust. This can easily be switched from your Gmail address to your new address later.
• Purchase a domain. Think hard on this, you want to keep this forever. Don't forget to check those renewal costs. Buy several years. Just forward it to your Gmail or other free email account at first, and start sharing it instead of your Gmail address (you won't be able to send with it yet).
• Look at switching to a mail client for your day-to-day email use. Every device has decent multi-account mail client available. Add Gmail to it and start getting used to not using Gmail's web interface anymore. This drastically reduces the amount of behavioural information you are sharing with Google.
• Switch your custom domain to a paid mail provider. Lots of options in this thread. You can change this again at any time, as you are using a custom domain. Connect it to your multi-account mail client on your devices alongside Gmail.
• Never delete your Gmail account -because then someone can scoop it up and impersonate you with that service or contact that doesn't know you switched. Delete all your old mail from their system, and unsubscribe and disconnect everything sending mail to it -remember about 2-factor authentication. Eventually you can take it out of your mail client so Google no longer sees your device activity - just log into the web interface at progressively longer intervals to verify you didn't miss anything important.

Good luck, and take it one step at a time. Perfect is the enemy of good and all that.

If only you or your relatives use the domain, it’s pretty easy to link all the addresses. It’s pretty fair to assume that if a search doesn’t give anything, the domain is probably personal or used by a few people at best

My point was that using a custom domain makes you less anonymous than being on the regular proton domain

If you do not use aliases, it won’t change much, but at least your identity is not directly tied to the domain name

You're completely right. I wrote privacy instead of anonymity

If you want to protect yourself from the ads industry an like having your own domain, then it's pretty good

The problem is being able to do this on all devices, and easily

But sadly with manual, we also lose the ability to do mail searching