For #1 (congestion), greater availability/convenience of taxis gets other private vehicles off the road. There might even be a greater than 1::1 tradeoff -- one more operating taxi eliminates more than one private trip. (Think of the time savings from not needing to park, for one.) Also, if congestion is the issue, capping just one contributor -- taxis -- while leaving other factors uncontrolled is crazy. If congestion is the concern, target a policy at congestion -- such as congestion-zone tolls, traffic-jam-taxes, etc.

For #2 (livable income), you nail it: medallions boost the income of medallion holders, not drivers in general.

For #3 (w.r.t 'public safety'), I'm surprised no one has pointed out: you can set minimum standards of equipment, service, and driver skill *without* capping supply. That's how most other regulated industries work.

For #4 (and part of #3, markets can't set rates), there are both low-tech and high-tech solutions to allowed informed rider decisions.

Low-tech (or rather, no tech beyond what's in taxis today): You could regulate the 'unit' of service -- a pickup is 3 units, each 0.2 miles is 1 unit, each 5 minutes in traffic is 1 unit, whatever -- then let every fleet or even cabbie set their own price per 'unit', a simple single number to compare. (And, that can vary by time-of-day, weather, etc.)

Or: you could cap the number of different pricing schemes at some smallish number (3-5), but let them vary completely by operator choice at occasional intervals, and require cabs to clearly indicate which scheme they use -- ideally by their color. Regular cab users (and tourist guide books) would quickly establish some rules of thumb making choosing cabs and knowing expected fares possible, but there would still be a mechanism for price variety and competition.

(High-tech, it could all be displayed/explained/calculated/negotiated/paid by your phone.)