Most places in the US, the taxi service is highly regulated not merely for safety but for the purpose of excluding competition. The common "regulation" is that a new taxi service must show customer demand that requires their new service. This is typically overseen by a board of their competitors who never seem to agree that they need another competitor.
You're spot on about the near impossibility of starting a new cab in a lot of major US cities. The price of a medallion for a cab in NYC is roughly $360k for an independent operator. They have a fixed number and nobody else is allowed to participate, regardless of need or service availability.
The concept of "everyone should play by the same rules" is pointless with some groups are grandfathered in and allowed to play.
Having read some discussions of Uber in Sweden, I think, regulation just meant a proper license, posting fares and insurance. All relatively reasonable requirements of a regular taxi service. There are issues with the posting of fares as Uber is a one-off service of unique trips, but that's perhaps an issue of updating the law. There is still the problem of traditional taxi services using the law to prevent a new service from coming into existence. They aren't truly interested in the customer who this law is supposed to support.