In the case of taxi dispatch, a car may never show up, and you won't know until you get tired of waiting, and there are no consequences for the taxi company or driver if they do this. Pre-Uber, in SF, the normal quoted wait from a taxi dispatcher was on the order of 20 minutes, which if anything was usually understated. If you're in a lot of pain you probably don't want to wait that long.

I would probably still call an ambulance for something I thought might be life-threatening, but for anything else I'd take a Lyft.