If the password is ever exposed via a breach, generally smart people can pick up on that pattern and then all of your passwords are cracked.

There are ways you could alter the pattern, maybe a separate short salt for each site in addition to the domain, so all you need to remember is the salt, like "dog" for pets dot com, that could make it a bit more secure. Or vary how you combine the master PW and the domain (i.e. count the number of letters in the domain and insert the domain starting with that number, or each letter of the domain that many characters appart embedded into the master PW.)

Or...just use random passwords, a password manager, and make sure you both trust the provider of the password manager (or use one you control) and use a super high security password as your master password in addition to other forms of authentication. And never let it allow you to stay logged in to your password manager.

I personally use a password manager except in a few physical cases where I use the function approach or have just directly memorized the PIN (iPhone, YubiKey pre-fingerprint).

The function approach is a normal password, just that it scales O(1) with your memory for O(N) passwords. The expectation is that if you have a complex enough function it will be non-trivial to back-fit it, this doesn’t necessarily hold for any arbitrary function, of course.