Thanks everyone!
@zenvelo You’re welcome!
I tested this out and it worked perfectly. There was some small damage to the photo, and I only sent it within the city. I simply had 4×6 photos printed, addressed them with Sharpie and put a stamp on it, no extra backing for thickness. I measured the thickness of the photo print (from Walgreens) using a digital caliper and it exceeded the minimum USPS limit. (USPS says on their website that index card thickness is acceptable.)
So all in all, I would do this again if I need to make a cheap and quick and local mass mailing as was the case here. If it were being sent a greater distance, I don’t know how it would hold up. I have never had a problem with mail getting wet, but the photo paper is not that durable and I think friction with other mail can lead to damage. There are a lot of websites that print “real” postcards inexpensively if you have the time to wait for it to be shipped.