Earlier quoted context omitted.
The problem with this solution is that it is only as strong as your master password. Because you suggest to 'never write or store [it] anywhere' it can't be strong enough. To say it in Bruce Schneier's words: "Pretty much anything that can be remembered can be cracked."[1] [1] https://boingboing.net/2014/02/25/choosing-a-secure-password...
I memorized 1024 digits of pi in high school. I can deal with a strong password. Keep in mind that most password managers also encrypt your password database with your master password, so my solution isn't any worse than those. Memorizing a even a 16-character (upper/lower + symbols) random string as your master password would be 16*6 = 96 bits of entropy which is more than enough. Dealing with memorizing ONE good 16…
Also, a character has at most 8 bits of entropy, not 64. If you use base 64 its only 6 bits of entropy. 16 x 6 = 96 bits is still more than enough though.
A downside if everyone used this scheme would be parallelized attacks on reversing the hash for the key. If you find a key that, with this scheme, creates a password for your service, you found the corresponding secret key. This then compromises all of that users passwords.