Do you really care that much what random passing-by drivers think of you? Are you that dependant on other people's opinion of you? I couldn't give a stuff.
I pay for NRMA Roadside Assistance, and I'm bloody well going to use it. They don't usually take more than half an hour to come, and if they say they will take longer, and I have a jack in the car, I might do it myself.
I change all four tyres and all brake pads to track rims and pads in about 40-45 minutes, so yes, I realise that I can probably change that flat in about 10 minutes flat, BUT........ I know how dirty I get doing it... My hands get FILTHY, and I need to use Solvol to get it off...
If I'm in a suit or about to go out, why should I do that to myself if I'm paying someone to do it?
It's like paying for insurance, but then panel beating your car yourself if you smash it.