"The knowledge" may very well at one point have been a reasonable requirement for a taxi driver, but at least as GPS satnavs got good, it's just not anymore. The purpose of "the knowledge" today is only to serve as a barrier to entry to protect incumbents against competition.
Edit: Footnote on doctors: It's a poor analogy as a taxi-driver taking a poor route (picking good routes is the singular advantage the knowledge provides) doesn't carry anywhere near as bad consequences for the "patient" as a doctor making a mistake.
It's kind of like if travel agents were regulated, and suddenly expedia or kayak existed. They can do an almost-as-good job through technology, and a "mistake" analogous to the taxi driver taking a poor route, i.e. Expedia connecting you through a worse city or not finding a deal 10% cheaper.
The same with self driving cars, its not like Uber or the taxi industry are long for this world without drastically changing business models (maybe 10 or 15 years tops).
It's sad but they are simply a victim of technology. Before sat nav, minicab drivers were very unreliable. They would get lost, they would have to consult maps, etc. Now, a minicab is as good or better than a taxi.
As usual it turns out that there are good & bad taxi drivers, and good & bad mini-cab drivers / companies too.
Part of being a good taxi-driver is knowing which parts of the city are going to busy at different points of time.
If a taxi takes a detour from the best map-presented route you don't know if they're giving you a faster journey, or ripping you off. Too often people assume the latter.
Google has all the information a cabbie has (road layout, major disruptions, historical traffic patterns, speed limits). They also have real-time traffic information. This is the kind of problem that computers tend to beat humans at, eventually.
Using the same kind of volunteers who contribute to OpenStreetMap it should be possible to update routing based on temporary road-works (which might last for 3-5 days, here in Scotland), which would otherwise catch drivers out by surprise with detours.
Assuming your conjectured startup somehow conferred the medical skills you would have just found a great way to lower everyone's medical bills.