How does zsh find commands?

zsh doesn't expand ~ within quotes. You need to use

PATH=~/bin:$PATH

instead.

PS: And path=$PATH is not required, zsh does this automatically

You do not list how exactly you checked that the path is actually at the beginning of $PATH - but generally you should be aware that macOS Catalina changed the default shell from bash to zsh - and as such you'll need to move over any modifications you've made to PATH in .bashrc (for example) to the corresponding file for zsh (for example .zshrc, .zshenv or .zprofile).

The "standard" way of adding paths to macOS is to create a file at /etc/paths.d/ containing the path to the directory you want to add to the path. Then when you launch any shell, it should automatically be added to your PATH environment variable. For example create a file called:

/etc/paths.d/wgroleau

And make it contain just the following line of text:

/Users/wgroleau/bin

This would system wide change default paths to include that directory.

UPDATE: Following your comments I'm starting to think that your problem might not be PATH-related at all. You state that the problems you experience revolve around scripts that you have written yourself. If those scripts happens to be bash-scripts that start of with a shebang for bash, then those scripts needs to be converted to zsh to work if you haven’t got bash installed - or alternatively you could install bash again from HomeBrew or from Apple.