The providers that I see almost always require a photo I.D., at least the first time I see them. I carry my drivers license and retired I.D. everywhere, but have never been asked/told to put one or both in an envelope. How about also getting a VIP membership here, reading the reviews and maybe they will help answer some of your questions.
jhb

You say you understand about the screening, so think about this:

So if I verify that John Smith is a (fill in the blank with respectable citizen work title) and thus not likely to be a threat; what good does that do me when you show up if I do not also then confirm that YOU are John Smith?

However, it's not often that I actually ask for the ID itself, those are too easy to fake.  When I do, I certainly don't want it in the envelope, it seems it might be too easy to forget to give it back to you and that can create a multitude of problems later...

Everyone has their own methods.  If a lady says that she wants to see your ID in her FAQ, you best be ready to show it to her when you arrive, or pay her for wasting her time.

.02

she asked to see my license to check if I were an organ donor.

Only once and with a copy of my flight itinerary to prove I wasn't local (and possible LE)

I bring it, but rarely does the lady ask. Maybe because I have good references. Even so, if I were the lady, I would check.

I heard from one lady that it doesn't matter. By the time the guy shows up, it is too late if he is LE.

I haven't been asked for my ID from any of the SPs Ive seen. Possibly having joined a verification service as I did helped bypass the need.

So it probably depends entirely on the lady. Best to get VIP, and read up on some reviews!

Otherwise, how does she know that you are the person she prescreened???

If you see well reviewed Ladies, your identity is safe  with her. She already knows who you are if you are being honest what dif does it make. She needs to feel comfortable and know you are who she screened.