In most single payer systems, yes, there are limits on what the payer (i.e. the government or other universal insurer) will pay, for precisely the reasons you state. However, the prices that are considered 'normal' for particular procedures and treatments are agreed upon and updated regularly by a board of generally independent medical professionals. These are people concerned mostly with the efficacy of the treatments and what the rules are for deciding whether they are medically necessary, rather than the financial aspects of the system per se. The government then uses these guidelines to make budgetary forecasts. So unless doctors are charging way-beyond-typical prices for things, it's fine. (And if they do, they'll be investigated and may be forced to eat the cost themselves, it can't legally be passed on to the patients).
However it's crucial to point out that if a procedure is medically necessary, it will be covered, regardless of how expensive it is. The government does not have unlimited funds, but the law of large numbers means that the costs over a whole population are quite predictable. Costs will increase over time as the population ages of course, which may require some tax increases or higher co-payments in future, sure, but this is still much more efficient than a non-single-payer system. The point of single payer is that a large customer (like a government) has bargaining power with providers and drug companies. They can negotiate cheaper prices for things than you or I could.
Everyone needs medical care at some point in their life. So it's not like insurance for things that may or may not occur (like car insurance or home insurance). Amortizing the costs across a huge population that uses their collective power as consumers to negotiate better prices with medical providers makes sense.
Yes technically this means you may be paying a few dollars more in tax to cover the tree-hitting motorcyclist or the drug addict or the VD dude. But that's a small price to pay for knowing that you will not have to worry at all about money if you yourself run into a tree the following day, or when you're old and are afflicted with cancer etc. Besides, the costs saved purely by the increased EFFICIENCY of having a single payer (i.e. much simpler, less paperwork, less billing, simpler IT systems etc.) I'd say more than makes up for that extra few bucks in tax.