echo 'export PATH="/usr/local/mysql/bin:$PATH"' >> ~/.bash_profile by using this command getting permission denied?
I would use the nano editor to edit the correct file for your shell, which is zsh, not bash.
nano ~/.zshrc

Put this line in at the top of the file
typeset -U path PATH

Put this line towards the bottom and save:
path+=/usr/local/mysql/bin

Press the control key and the X key to exit (see the bottom of the nano window for more commands). If you then quit your terminal app, you can test for mysql being in your path.
You can tell your shell with echo $SHELL and if you want bash since all your instructions are in it, you can go back to that if needed with chsh /bin/bash in which case you would nano the file ~/.bash_profile since each shell reads different start up files. I recommend learning zsh and finding guides that work there, though.