ZSH takes two tabs to complete filenames

You can add your completer to the list of completers used by default:

zstyle ':completion:*' completer _last_command_args _complete

_kadmin is probably a completer function for the kadmin tool - not a directory. If you attempt completion on something that zsh can't find as a command, a directory or a valid and known command argument completion, it then starts to offer completion functions as possible expansion candidates. By default, zsh comes with a lot of completers, many of which you may not need - there are bundles for AIX, BSD, Cygwin, various Linux distributions, etc, and they all get read and installed into the shell. If you attempt an expansion on something zsh can't find, it has all those installed completion functions to offer you instead.

You configure zsh not to offer completer functions by putting this in your ~/.zshrc:

zstyle ':completion:*:functions' ignored-patterns '_*'

Reload the file and you should no longer be offered completion functions for tools you don't have installed. Have a look at the zshcompsys manpage for (a lot) more detail.

EDIT in reply to UPDATE 3

If _kadmin is actually a user account, you can configure zsh to not offer it in completions. It seems the approach is to list the user accounts you do want the shell to consider, which limits any names offered only to those listed. The zstyle line is something like this:

zstyle ':completion:*' users asgeo1 root

I think you can list as many users as you like after the users tag. The shell will then only offer those users' home directories as possible completions for the cd function or builtin.

I don't know why adding the username to the ignored-patterns in the completion.zsh file didn't work - did you reload your config after making the change?