That works until you visit the website for my insurance company...

"You must use one of these special characters: %^&*()"

How does incrementing a counter address some sites requiring special characters while others reject them?

The counter only addresses requirements to change password periodically. The special character requirement is addressed by tacking on 'Aa$1' to all passwords (they asked for it; it's what I tacked to all the (good) passwords I memorized even before I had this method).

All your passwords have special characters. Some places do not accept special characters in passwords.

This is what your scheme doesn't address.