A new breaker would be important, given that the existing installation is a pair of single pole (120 volt) breakers, which is illegal and a life safety issue. Might be a good idea to ask the electrician to review the panel and the house in general, in case there are any other serious issues.
Perhaps I haven't described it well--the electrician said that the panel was quite well done. He replaced the breaker and also relocated the dryer to a better spot. It was more expensive than I thought, but I'm glad I didn't try to do it myself. We can use the cooktop now while we wait for the part to fix the oven. Thanks for the warning!
No, you described it just fine, as far as I can tell. A double breaker covers both incoming bus bars (the copper plated metal that the breaker clamps on to) Each bar supplies 120 volts and, in combination, they add up to 240 volts, which is required of most ranges, electric dryers, well pumps, electric baseboard heat, and other heavy amperage draw loads. If you have the ability to switch each side of a double breaker off and on, you have an installation where somebody used two single breakers, side by side, to supply 240 volts. That is, as I stated, illegal and dangerous. "Back in the day" this was an acceptable practice, since breakers had a hole drilled through the end of the toggle switch, and you could, and often did, cut an eight penny nail and drop it between two toggles to "make" a 240 volt breaker, by tying them together.
It sounds like your electrician resolved the issue, and is confident that everything else looks good, which is good to hear. The electrician corrected a potentially dangerous situation that occurs when an unqualified person decides to repair the stove, and heads to the panel to turn it off. They look at the panel labels and determine that the breaker for the stove is breaker #8. They turn #8 off, and tear into the stove. Unfortunately, the stove is still fed with 120volts by #10, and they get electrocuted. Now the event is a preventable chain of incompetent practices, with a lot of opportunities for a trained and qualified person to have prevented the tragedy, but codes and best practice are there to make everything as "idiot proof" as possible.