This may be stupid, but can you run regular ni-mi batteries with a brushless motor? Will it be just as fast as the brushed motor or faster?

not such a stupid question
brushless motors can be run on nimh or nicad,lead acid,lithium polymer,lithium ion,lithium magnesium and probably a few others.
depending on what brushless motor your looking at will depend on whether it will be faster than a brushed and depends on the brushed motor.
if your looking at replacing a standard type 540 can with say a sv27 motor or similar yes it will be faster
most types of common setups like in rtr,sv27, ul-1, blackjack etc will run on nimh but the motors are fairly hard on these batteries as they cant deliver the amp draw that a typical
lipo can,and they get very hot,but saying that it wasnt long ago that lipos were not used and nimh powered some very fast setups.
if you supp-lie what motor your looking to replace and in what boat and batteries etc someone should be able to give you a idea or what you need

Another way to think of it is that the motor (or actually the speed controler) requires direct current, a constant flow of electrons. What chemical reaction provides it is not important. The volume of current required may dictate which chemistry is necessary but in the end the ESC takes direct current and depending on what type of motor is being used turns it into a usable flow of power (current).