Actually, never mind. I just found a webtool that will do what I want.
https://www.gradient-scanner.com/
Just drop in an image file and mark where you want the gradient scanned. After that you adjust the sensitivity and it spits out the stats. I still had to type them in by hand but it is VERY accurate. The location of each color is listed by 10th of a percent and PS will only accept integers so you’ll have to round them off.
Here is a sunset (or sunrise, I’m not sure) that I ran a test on. The left side is the original JPG and the right is the gradient I was able to create.

The only adjustment I would want to make would be to tighten up a few of the Color Midpoints. If I were willing to futz with it for a while I could get it closer but considering it only took me 4 minutes to create a 15 Color Stop gradient with this much accuracy I’d judge it a success.
If anyone can find a more automated method that would be nice (something that actually spits out a gradient file), but if you need better accuracy than eyeballing it with the eyedropper tool this is pretty good.