You're seeing problems that aren't really there.
> All businesses must be licenced.
Why? Why the hell would a grocery store need a license and what for? I'm not asking for "because the government says so", because the government says a lot of things that don't actually make sense. Is there a good, objective reason for all businesses to have to be licensed and how would the license help?
> Taxi licensing is special because of their special circumstance.
What does that mean? What are the special circumstances? It's just people with cars and people who need to go places.
> (a) 'forget' about picking up people of a certain skin color
Suppose there was no regulation for taxis. Now suppose a big taxi company had emerged. Let's call it "Toober". Now Toober, just like most companies, wants to make as much money as possible.
Do you think people accusing its drivers of racism would be conducive to that goal, and if not, do you think Toober would do something about racist drivers?
What about the drivers? They want money too, right? So if there was a racist driver working for Toober, and he knew Toober doesn't tolerate racism because it's bad for their image and thus, bottom line, don't you think the driver would refrain from behaving in a racist way so that he could keep his job?
> (b) refuse fare to a certain district they don't like
Don't like? Because it somehow displeases them like Justin Bieber's music displeases a lot of guys? What sense would it make for a taxi to leave money on the table because he finds a district distasteful?
> (c) arbitrarily set fares depending on what they see as people's willingness to pay--like say setting the fare for single women outside nightclubs to 3x the normal rate.
Again, imagine Toober getting complaints about discrimination. On the other hand, getting home safely from a night out is a more valuable service than just getting from Place A to B in less dangerous situation, so maybe it's alright to charge a bit more.
Sure, three times the normal rate would be too much, but another taxi company would be free to offer a better rate, and people would take it.
Do you see why regulation is not necessary at all? Whenever the government says it's doing something for your safety or for fairness, you can be 100% sure the results and real reasons are different.
The real reason why we have taxi regulations is to maintain a state-supported taxi-cartel.