If you are going to place a crown on this tooth and the old root canal has been exposed to the bacteria in your mouth(cotton ball had nothing to do with it) then it would behoove you to make sure the rct is in good order. I would suggest you have the rct retreated and the crown placed.
Your dentist placed a cotton ball under the temporary filling to make it easier to remove the temporary when the core and crown preparation were to be done. Endodontists have advised that even a week of exposure to the oral cavity can be enough to allow bacteria to compromise the root canal seal possibly leading to failure.
Depending on when your temporary filling began leaking, I would say it's highly likely that the root canal has been exposed for some time. As the other dentist mentioned, the cotton ball has nothing to do with much. It doesn't generally affect the temporary leaking over time or have an impact on the outcome so whether you knew about it or not shouldn't matter.
Ideally the root canal should be retreated, but depending on what you want to do, how much of a gamble you want to take, and whether the dentist would be willing to do it - you could try to have the core/crown placed anyways as long as there are no radiographic signs of an abscess or symptoms. I would recommend that the dentist remove a bit of the coronal root canal filling material to attempt to rid any fluid / bacteria that may have seeped in.
Some newer bonded sealers (eg. Resilon / Realseal) may be more resistant to bacterial invasion. I'm not totally up on the research on resistance to coronal leakage with these materials so take that with a grain of salt, and also I don't know if it was used in your case.
The problem with not retreating the root canal is that if you go ahead with the cost of the core and crown, and if the root canal fails at a later date, you likely would have to have a hole cut into the new crown/core to have the root canal retreated. This could mean that you would need to pay for another core/crown.
Sometimes due to a patient's cost contraints, we can't follow ideal dentistry. As long as you understand the risks, it's up to you if you want to try what I mentioned. Anecdotally, I have seen some root canals that were exposed to oral fluids for months survive just fine with no problems. The patients were informed of what ideally should be done, but they decided to accept the risks in not retreating before the restoration. Call it non-ideal or substandard treatment if you will, but it is at least an option.