Another thing to note. Your spare can probably be inflated, but the question is can your pump inflate the spare.
My pump was "working", but it definitely was not inflating my spare. We were in the middle of a track and had to get a ride to the garages with my spare and luckily we found an air compressor to fill it up. Although I have used my pump in the past to top of the pressure in my tires when I first bought the car and couldn't find a gas station with an air compressor. But the folding spare takes quite a bit of pressure to unfold and inflate.
I don't carry the spare with me anymore and definitely don't really worry about it around town. I did put it back in for my last major road trip to NSXPO last year at the insistence of my girlfriend.
Another scenario where having my spare didn't make a difference. After some mountain driving, on the way home I discovered both rear tires had become concave from the internal belts shifting. So at speed the car was experiencing very rough handling. Using the spare would've solved one side, but the other side would've still been an issue. I was luckily able to crawl my way home. Again this is sort of a worst case scenario.
Now, I just try and keep track of my tires' condition and hope for no random punctures.
Definitely do try and inflate your spare to see if it will hold air though. And give your air pump a test too.