I don’t buy online because I want to see how they look on my face first.

Many shops won’t sell you glasses unless your prescription is under 1 year old. This allows them to charge you for an exam when you don’t necessarily need it.

Sites like Warby Parker will send you a bunch of frames to try on for free so you can see how they look.

And I find sites like Zenni are so cheap that it's worth taking the risk: an ugly pair can go in the car /gym bag / work desk for if I lose a contact lens.

Every time I change glasses I try at least 20 frames, more likely 40 frames, before finding anything close to acceptable. Something that looks fine on a shelf most often looks ridiculous on my face. Can't imagine ordering online.

Lots of Zenni frames are so cheap that it's non-crazy to just order like 3 pair you think might work, and keep the two you like the least as backups, wearing the one that looked best. They also have "see it on your face" tech that's... OK, for narrowing it down. Better than nothing, and better than looking at them on a model.

What about the privacy issues of sending them my face for them to keep in a database forever?

Some of the online shops now actually allow you to upload a selfie, and they'll composite your chosen frames onto your face.

Probably not quite as good as a live view, but not bad.

What about the privacy issues of sending them my face for them to keep in a database forever?

I'm pretty sure that ship has sailed. Unless you stay indoors all the time, or maybe you live in Bush Alaska or something, your face is likely being captured and stuck in a database tens of times every single day.