Does the battery actually list it's A rating (not Ah) specifically, as a continuous and/or peak/burst rating? If not, the "watt" ratings aren't necessarily "safe" to use to assume what the battery is really capable of.
Also, since it's aged, that will decrease it's capacity and it's current capability by an unknown amount.
If you're not sure, the present capability can be guesstimated fairly easily with a wattmeter installed between battery and controller, over time as you ride, as you see how much voltage sag you get under how much current load, and how many wh/mile (or wh/km) it uses up (which you can either calculate from distance vs wh used, or directly measure with some wattmeters like the Cycle Analyst that also monitor speed/distance).
What specific needs or desires, other than those in the post, do you have that indicate a new controller would fill them?
(because depending on the needs, the controller itself may not do that, at least not by itself).
What specifically do you want out of the display, in detail?
(because the display is more or less just a display, and the core functionality from it has to be in the controller itself, but the display has to be compatible with it, so if you want features a display states it has but the controller chosen doesn't have those, you won't get them, or the display won't do anything at all but turn the system on and off, etc).
The controller doesn't achieve distance; your battery does that. If your present range is insufficient, you would have to get a proportionally higher capacity battery to change that.
Getting a higher current controller will *shorten* your range, at least proportional to the difference in current vs the amount of time the system experiences that higher current, and possibly worse depending on the condition of the battery (how much voltage sag it gets at that current (load) vs now, as all that voltage-sag x current's power is wasted as heat inside the battery, never making it to drive your wheel.
To know what current limit to choose for the controller, assuming you'd be willing to replace anything else that can't handle it if that turns out to be the case, you need to know the specifics of your hills, and what you want to happen on them. At the basics, power needed depends on the weight being pushed up what slope at what speed. It also depends on wind/air resistance, so if it's windy it will change the power requirements. Same is true of road conditions and tire pressure (rolling resistance), so if it's perfect roads and ideal tire pressure from the label, it'll be less power than with low pressure and bumpy or gravelly or sandy / etc roads.
https://a.aliexpress.com/_EjG5obr
Torque is proportional to current, assuming no voltage sag. So ideally, if you go from 20 to 30A, you'd increase torque by that same 3:2 ratio. Since the battery will almost certain sag in voltage by some greater amount under the higher load, the total power you get won't be as much, so the torque increase won't be as much.
Square to sine just quiets the system by some amount. Depending on the controller and motor design it may also reduce the torque some amount.
Using FOC (also sine but in a more advanced way) controller gives direct control over how much current the motor gets, so you control the motor's torque instead of it's speed. This changes the feel of the system and can feel more natural once you get used to the change.
Efficiency will be more affected by your motor's match to your specific usage than most other factors; there's more power wasted as heat inside it than anything else in the system.
If you're curious, you can experiment with the simulators at ebikes.ca to see how various factors affect efficiency and system performance.
