I went through all the cmd.exe steps and everything and it said that the drive wasn't ready, so I'm using the HP Support Assistant program to check the optical disc drive.
It's telling me that the media supported by my drive are: DVD-R DL Sequential, DVD-R DL, DVD+R DL, DVD+R, DVD+RW, DVD-RW Sequential, DVD-RW, DVD-RAM, DVD-R, DVD-ROM, CD-RW, CD-R, and CD-ROM.
As I understand it, the problem is mainly that movies are considered DVD-Video discs, which isn't listed as media supported my drive--but these exact same discs were supported without fail for two and a half years straight, so would replacing the optical drive actually fix that problem? Additionally, I just tried to do a read-only test using a commercial audio CD (produced in 2009) that I know for a fact my drive recognizes and plays normally, but HP Support Assistant tells me that it's a blank disc and the packaging doesn't list what kind of CD it is.
ETA: none of these discs are scratched and they all play perfectly fine in two other computers, so the condition of the discs is not the problem.