Bklein said:
I received a new controller
Is it an identical controller, or a different kind? If it's different, what are the specific differences? (old brand/model vs new brand/model, etc)
The new controller has a brown-black-yellow connector left open. I am assuming it is for the pedal sense, which on the old controller is red-green-yellow.
It could be for a lot of possible things, depending on the controller design. I don't know what that specific color combination might be. I wish there were standards for them, but while there are commonly used combinations, they aren't always used.

I have a handful of PAS-capable controllers, and three of them use red-green-black (like a throttle), another uses red-yellow-black. I forgot what the other one is.
Without a wiring diagram for the controller, which should be provided by the seller, either with the controller or on the webpage you bought it from, the only way to find out what the connections are is to open it up and see if there are interpretable markings on the circuit board for where the wires go.
Hopefully you can get the diagram and wont' have to open it up (markings are often similar but not identical, and may not be interpretable with certainty).
The only other thing is: are my expectations correct? I switch on the battery, press the control on-off button, select a mode 1 2 or 3, and turn the throttle and it should go?
That depends on the specific controller design. Simple ones do work that way, but others don't. Some possible examples:
If you have a controller that requires PAS to be engaged before it will allow throttle usage, then you'd have to hook up the correct kind of PAS sensor to it (there are several), and be pedalling, before the throttle will work.
If you ahve a controller that requires a compatible display/controls to be connected to turn it on and to change the mode from zero (no motor operation) to one that allows throttle-only control, then you'd need that specific display, etc. Not all displays are compatible with all controllers; there are at least several communication standards, even when the wiring and plug are the same. If you have a controller that requires the display, it's best to get them as a set to be sure they'll work together.
It could also just be that the controller has a keyswitch/ignition wire (often a single thin red or orange one bundled with the thicker battery plus and minus wires), that needs to be connected to the battery plus to enable the controller.
Etc. Any info you can give about the new controller, like a link to the seller's page, etc., may help us help you find why it's not working.