The sole reason the website broke is because there was a massive upsurge in people accessing it, well beyond the normal rate for tax renewals as people were for some reason waiting for it.
I've done my previous 8 vehicle tax renewals online via the DVLA website just fine (yes, this isn't the first time you could buy your vehicle tax online, they've had it for years, all they are doing now is not sending you a physical tax disc) and the website has been fine - in this case I wouldn't lay all the blame on the service provider as they were working to previous usage levels that have been long established.
As for health care services, well I've never had to fill out a form relating to health care in the UK, I just receive the care that I need. Oh, and I can book appointments, order prescription renewals and even choose a specific doctor to have an operation with online. Have done for years
In summary, the system isn't as broken as the story makes out.