OK, let me put my experience on this as well.
One day, I grabbed the wrong spray bottle out of the prepaired totes we used for detailing. Each tote was color coordinated for their area (exterior/interior/cloth tops/engine/wheels) you get the idea.
Anyway, one of the guys put the spray we used for leather cleaning (Woolite/water mix) in the clay tote.
Anyway, I grabbed it spray it and started to clay and right away, I noticed the lack of smell from the normal lube and started to freak.

What did I just spray on the finish?

Oh crap, so opened the bottle and lucky for me I then realized it was Woolite/water. Interesting... It did lube well, and was not worried, so I continued on the fender. Rinsed it off and checked it. No poblems here!
So I gave it another try on a door, and it seemd to work rather well.
We laughed about it and thought, what the hell? Can't be all that bad? So after finishing a whole car and rinse, didn't see any issues. Besides, finish was going to be polished anyway.
So in reflection, I did not see any issue using it, works well for a lube and can cut it in different ways to provide more lube w/o gumming up clays. I also used another small bucket to rinse the clay between areas and it also seems to remove dirt off the clay and the larger particles were firmly in the clay as before which required the clay to be folded or cut.
Although I did not try it with ALL clays, for which ones we used, it seems to work.
Also, one of the guys used the Woolite mix for a cooling spray when polishing a panel with some polishes, but where it really worked well was on plastic polishing. It lubed the pads and polish so you could work longer and let the polish break down. Only water based polishes worked, any polish that does not use a water based oil, or emusifier turned it into a mess and cleaning the pad was even worse, so lesson learned.
Bottom line, Woolite is not going to harm a finish. It's a soap with a neutral PH and so far has not done any harm to any of the cars we did in the past.
Someone learning or new to claying a finish, I'd go the normal route since most new users have no idea on correct technique or have experience and the lubes made for clay have more lube as a forgiving cushion.
Just my experience in this matter.
Regards,
Deanski