Sometimes it seems like the wild wild west out there the way a lot of small non-profits (and churches in particular) operate, and trying to explain the rules and get them to do things the right way as opposed to the easy way and the way they've always done it can be a big up hill battle - and it's just a lack of understanding from volunteers who take the path of least resistance.
I don't think you are wrong. Administering payroll for someone else's employee(s) - not a problem, I guess, although who is listed as the employer (EIN, etc.) on the W-2? Considering that person as your employee for benefit purposes, especially retirement plan(s)? Big problem, as I see it, and would definitely dig in your heels to do it right (or have the other employer adopt as participating employer).