My team has set up continuous servos that move as wanted. However, they don't brake (they just become limp at the deadband values). Is there any way to make continuous rotation servos brake like the standard servos do?

Continuous rotation servos do not have built in position detection. The potentiometer that is in regular servos has been removed so that these servos can continuously rotate. Thus, there is no way to have them stop at a specific value unless you add an encoder or potentiometer to the axle that the servo is controlling. If you are fine with doing this, then you can write code to have the servo "brake".

I always don't understand continuous servo. Aside from motor power, form factor, different motor controller etc, in terms of functionality, if I have a situation that requires a continuous servo, why not use a regular motor?
We have an elevator on our robot, our mechanical team wanted to use a continuous servo at first but the programming team wanted to know the exact height of the elevator and of course be able to hold the height. So we decided to use a tetrix motor with an encoder. But then I can't really see what scenarios that must require a continuous servo that a regular motor cannot do.
The only scenario I can see is if you just want to turn something continuously and you are running out of motor controllers (since motor controllers can only take two motors but servo controllers can take 6).
Our robot also has a "conveyor" that picks up racquet balls that we just need it to keep running or stop. So it sounds ideal to use continuous servo. But we ended up using a tetrix motor on it too because we were worrying the continuous servo may not have enough power. So we really can't find too many scenarios that continuous servo is a must.

Continuous rotation servos may be used when a motor is too heavy for the application, and it is not possible to bring motor power to the required spot on the robot using chain or linkages. That's pretty much the only reason I can think of besides the ones that you pointed out to use a servo in the FTC game. However, continuous rotation servos are great for hobby applications. They have basically the same functionality as a motor, and come with a motor controller and (usually) power regulator built in. All you have to do is supply power and send a pulse. This is why until last year, all VEX motors were actually continuous rotation servos.