With 6A you are at a very high current.
First about the batteries: I wouldn't trust in 4 AAA batteries in supporting that high of a current. You may get away with it, if you only have very short bursts of 6A, but I guess the voltage will take a significant dip here. I would look into LiIon or LiPo battery packs, like the ones used for drones (which also can have high current spikes). But please take care to also use a fitting charging circuit. You can find many resources about that on the web.
Now about the breadboard: Just don't pull 6A through a breadboard! Breadboards are not made for currents this high. The contacts have a resistance that is significant in this situation and the metal strips used in there are also not meant for carrying that much. Going over 1A is probably no good for a breadboard. Instead you should connect the power supply outside of the breadboard by soldering or using proper connectors, which can handle the current. You can hook up your breadboard power regulator to the same battery if you want to provide it from the same source.
Where can I find a power supply that is arduino compatible and outputs ~6V while having a max current draw of 6A or over?
There is no "Arduino compatibility" with power supplys. For the Arduino its just about having the correct voltage and being able to provide the needed current to the Arduino (excluding the servo, since it shouldn't be plugged into the breadboard in the first place). When you use a battery and also want to charge it in your circuit, then you need to find a charger board compatible with your chosen battery. Often these boards also have a voltage regulator from which you can get the 5V that you need to power an Arduino. Otherwise you can buy voltage regulator boards, which can take a full range of input voltages and output a regulated voltage of your liking (controlled via a small potentiometer on the board). These can also be used if you want to use a power supply connected to mains (like the often seen 12V power supplies for LED strips).
So called "breadboard power supply modules" are often not more than a board with pins in positions fitting for a breadboard. They are a pure convenience product. You can also solder pin to the cable of power source and put these into the breadboard. Sometimes these boards also have a voltage regulator, which you can also get without the breadboard connector part.
