First, I'm going to use the term "taxi service" below and I'd like to clarify that I mean this in a very general sense - paid transportation in vehicle classes that would normally be used for private transport, whether provided by a firm or individual, and facilitated by any means. I don't think we disagree based on the tone of your post, but this is for other potential readers. Please let me know if you do disagree.
In this case taxi service need not be supplied by full-time taxi drivers, which is a central point to most of my thinking on this matter. Taxi service can be provided in any quantity (subject to the limits of time and space, obviously) by any individual with a car. A person may provide taxi service once in their life or dozens of times a day.
Second, I will refer to Uber below for simplicity, but I intend my argument to apply to all similar services that exist (Lyft so far as I know) and may come into existence as competitors.
In the status quo the number of medallions is a rough approximation of the limit of taxi services. I think we can agree that the current number of medallions currently limits the supply of taxi services (the supply curve is shifting toward the origin because cost of entry is higher than in the complete absence of medallions or equivalents), and therefore raises the equilibrium price of taxi services. Currently the medallion represents an economic rent (regardless of who owns the medallion a rent is being collected).
In the future state (this is a very general future state - anything where the current limit imposed by medallions is weakened or removed entirely and at least some of the rent that exists in the status quo is transformed into revenue for Uber) the situation is improved. The revenue collected by Uber is not an economic rent, but rather a transactional cost.
First, the cost structure is different - a transactional cost is marginal, whereas the medallion rent (this is specific to a medallion, not all economic rents in general) is either not marginal or marginal at a much coarser granularity. If I rent a medallion on a monthly basis, then I am deciding whether it is worth it to me to drive a taxi for another month. With a transactional cost, I decide whether it is worth it to me to provide one more trip of taxi service. This allows someone who might just pick up people on their normal commute (this is just an example that I find easy to discuss, not the entire basis of my argument) to provide taxi service. Right here we see an increase in supply of taxi services. Our first order analysis should lead us to expect that this increased supply should lead to downward pressure on the cost of taxi service.
Second, the transactional cost is incurred by a competitive firm (currently one competitor in Lyft (that I know of), but even nonexistent competitors can provide pressure[0]). If a competitor can provide a similar service to Uber (matching willing drivers to willing passengers) at a lower cost, then the transactional cost will be driven downward and since there is clear competition in this case among drivers, the cost to end users will ultimately be driven down[1]. In this case, even if we accept your equivalence of Uber's profits and the current economic rent of medallions (note: I do not accept this equivalence, which I think the rest of my post makes clear) the $/ride of this money will decrease, resulting in a net gain.
I have not yet addressed your points about insurance, but first I would like to conclude my current line of reasoning by addressing your fourth paragraph.
You discuss driver incomes and wages, with a comparison to minimum wage. I do not believe the connotation of these terms is in alignment with what is made possible by the service Uber offers in matching casual drivers with occupants. I believe the innovation of Uber is just a re-realization (with some obvious improvements) of the jitney service of the early 1900s. I discuss this concept in more detail in another post[2] with a link to a very interesting history of the trend[3]. In brief summary, ride-sharing is a distributed casual endeavor. A useful metaphor may be Folding@Home - a service that utilizes excess capacity of capital with a significant sunk cost component. I do not think it is accurate to talk about someone's income or wage for something that is not their job. It is not strictly false to do so, since the terms are technically general, but they carry the connotation of employment and I do not think we face a model of employment. We don't talk about someone's wage when they host AirBnB guests. We do not talk about the average person's Craigslist furniture income. My position is that we should view ride-share as analogous to these two examples, rather than as analogous to the current status quo of taxi driver as occupation.
Finally, with regards to insurance, Uber provides insurance to its drivers which is primary to their personal auto insurance[4]. I do not believe your statements regarding insurance are factual or I am misunderstanding your reasoning. I would also like to understand why you expect accident rates and insurance claims to increase. I can think of numerous lines of reasoning, but I'd like to understand yours.
[0] Pressure from potential entrants can force a firm to act as if it had competitors. This idea is not strictly necessary for my argument, so I have removed its exposition. Even if you do not accept this line of reasoning my argument stands since there is no clear monopoly in providing services similar to Uber. So long as a competitor may enter a market, an incumbent firm faces competitive pressure. To wit, if Uber were the only firm in the market, there is currently no large barrier preventing a competitor from creating a similar service and offering a lower price. (There are costs to entry which provide some barrier, but cost of entry does not prevent all competition - there is a nuance that I'll not delve into here about capturing monopoly rents in proportion to the natural cost of entry to the market - it alters but does not destroy the line of reasoning presented.) If Uber wishes to maintain its singular position in market, it cannot sustain monopoly pricing. To do so, it would have to restrict its own supply to raise prices. Since prices would be above equilibrium, competitors would have incentive to enter the market an collect a portion of that monopoly rent. Upon entry of a competitor, we naturally see competitive behavior. If Uber wishes to maintain its singular market position, it must act as if it has competitors, else it will gain competitors. This process would keep prices in line. Monopoly rents are only regularly collected by regulated industries. We could argue causality there, but that is a different conversation and this post is plenty long.
[1] I suggest that competition among drivers would lower the price to consumers. This is not exactly true, since as we both know Uber sets the price, collects the payment, and pays the driver. What I actually mean is that since in the future state there is no artificial limit on the number of drivers, the natural consequence is for price to consumers to fall. Uber wants more customers, so they will lower the cost to their customers. There is an equilibrium to be found among driver remuneration and customer cost. I think we both agree that the trend would be downward for driver pay.
[2] https://news.ycombinator.com/item?id=8875663
[3] http://www.jstor.org/discover/724795?sid=21105612783593&uid=...
[4] http://blog.uber.com/ridesharinginsurance