If there were more taxis again in Paris, there would be some competition, price would go down and quality of service increase.
Reality disagrees with your assertion.
Quality is the first thing that goes in a race to the bottom and that's exactly what you're proposing. I've lived in places where there was an oversupply of taxi drivers. Eventually you reach a point where there are so many taxis they start to resort to any number of dirty tricks. These range from grabbing you and manhandling you into a cab to dodgy meters to forming gangs and enforcing turf.
One thing that is consistent is that the quality of driver is very, very low. This is consistent from my experience with Phuket's Tuk Tuk drivers to the Dominican Republics Motoconchos to Filipino trike drivers. Quality always suffers when its a free for all. Its like saying that a buffet will be better quality than made to order because it's cheaper.
If it wasn't profitable to be a taxi as there could be more supply than there's demand, people would switch to a different job and it would regulate automatically.
Every place with deregulated taxis disproves this assertion. The system does not regulate automatically, if the government does not regulate it, someone else will. Typically when regulation is left to the drivers they form gangs, establish turf and are pretty violent in enforcing it. Its not the clean, rational system you imagine.
Also, self regulation leads to higher prices, not lower prices. A taxi in well regulated Bangkok is very cheap, 400 Baht from the airport to the city centre. A taxi in unregulated Phuket is extremely expensive, they wont turn on the engine for less than 200 baht.