How do guys feel about FaceTime verification? I haven’t done it yet. I’m concerned a screenshot could be taken and my face would saved on a phone.

in which world you are playing.. I do it all the time in SB world - actually is one of my steps in vetting that I don't skip.. In P4P world I don't.. I have done FaceTime chat with provider because I wanted to ask about something that was too much to type, but it wasn't a verification.  Its not really a common practice in P4P but nothing should stop you from doing it should you want to.

Any woman who uses that alone for her screening is foolish.

If you are worried about someone capturing your face (not to mention other specific body parts) then why would you ever agree to meet a provider face to face at all?  You have no guarantee or assurances of privacy throughout the entire process:  
- Public street intersection cameras
- Hotel security cameras
- Ring doorbell cameras
- Hidden cameras in a provider's living room, bathroom, kitchen, parking lot
- Your cell phone pinging on nearby cell towers
- Random people recording on their phones for tourism, YouTube, or other purposes.  

Being captured on a public street camera, or by some rando on vacation, is quite different than doing a FaceTime convo in an illegal business. This little corner of the world of ours is filled with paranoia... and thats because some people really are out to get us. lol.
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To the OP, I would never do them as they are well outside of my comfort level. Much like HolyStone, I have been asked to do a short phone chat every now and again, but I would decline a FT request as I find that to be WAY too intrusive.

My purpose here is to point out that there is no such thing as zero risk.  Risk exists in everything we do.  

I haven't ever seen anyone make the opposing argument so I am bit baffled that you would state something so obvious.  
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And no, a FT chat cannot be as "sanitized" as a text. They are inherently different, as a FT includes video showing faces and can be copied, possibly for nefarious purposes, something text does not.  
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Your last paragraph was really puzzling. You state, "But let's be careful not to cast a blanket suggestion that ANY provider who feels she can be safer with a facetime chat is a scammer out to blackmail would-be clients." I have to ask...who is saying that???? Certainly not me, and furthermore, absolutely no one else in this thread, that is for sure.
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I like your posts, and generally agree with your views, but in this thread you put up several straw man arguments to counter points that literally no one here is making.

It just makes doxing easier for those that want to scam you.  A simple screen shot during the exchange is all that is needed.  

"it also provides some level of comfort that each party is who they say they are"

While I have never done it, I am not against it. I think that it can provide a decent chemistry check.

lol I hate it, and wish the concept would go away. Facetime is NOT screening. You should absolutely be cautious because they can blackmail you as the client with a live video. From the provider standpoint, most guys that ask for it are just trying to get a free peepshow. I believe it was made up by ex-blade girls that don’t know how to screen properly. They say “show me ur pp” and think that’s sufficient

Especially stay clear of the ones who make it mandatory along with the deposits.

...thought it was different, but then again, I'm still new and a bit naive I suppose.

I've seen ads on Tryst that offer the service of Facetime/Video Chat for a fee.

One of the scams on Tryst is the prepaid  Video  Chat  
Too bad many are for a session that was never going to happen.  
Only do with provider with a reputation here or elsewhere that has something to lose .

I knew a provider who maintained that video screening would eliminate LEOs. An essential part of her screening was a requirement to show your junk.

but I'm sure that if LE thought it could lead to a clean bust, they would show their (or someone's, perhaps AI generated) "junk".

I think it might  be useful.  But need to be careful.  One provider told me she realized after a Zoom verification that her real name was attached to her Zoom.  Be careful!

True but, sadly, a lot of providers still cling to these incorrect beliefs.

Yes, and I'm more concerned about the video-audio being used to train an AI model to impersonate me and to say things that are embarrassing, ruinous, and/or illegal unless I make blackmail/extortion payments.