But this got me thinking about my spare tire. It's never been used since I've had the car in 1996, but it's damn near 20 years old now. It may be time to look for a replacement.

. Maybe even a 17"er to match the other 4.....
off topic from OP's post but FWIW:
my "spare" tire on all of my cars is the "original;"...except the one in the SS.
years ago there was a class action suit, which I opted "in" for, that was won for the plaintiffs. It said GM stated on the window sticker of the "Impala SS" that it came with a "full size" spare. Compared to the size of the tires on the car (255/50 x 17) the spare was not. It was a standard 15" tire found on Caprice, Roadmaster B-Bodys. The settlement was $250.
So I got a 17" SS rim for $125 (new at the time) and a new 255/50 x 17 tire
The spare in the B-body is in the trunk and covered so it should be "useable" longer than one on the car or exposed to sunlight. IMHO "spare" tires should be used short term to replace a flat and put back as a spare once that tire is fixed or replaced....unless it is included as a "5 tire" rotation practice.
To the OP's issue with original tires.....they have expired in terms of their performance rating and should be replaced if driving on public roads for the safety of the occupants and others on the road.
I got a very low mileage but 7 year old Toyota from a family member I gave to my daughter when she went to college. Tires were original and had a lot of "tread" left. There were "cracks (small)" in the tires and I didn't think much of them. Car was parked outside its entire life. My daughter got a flat (puncture) and had it fixed at a local tire repair near her college. A week later she text me and says "my tire has a bulge in it (with a pic)"......I bought 4 new tires. Soooo tires do "expire" before the tread is gone if older than 7 years....so 17+ year old tires, even garaged, IMHO have degraded beyond their safe operating ability