Why throttle hard to turn? - 2006 Honda VTX 1300 S

Not sure how you put a battery in your AC, first time I've ever heard of a car AC having a battery.
Your AC may need "recharging", but that has nothing to do with a battery. A/C systems are filled with a liquid/gas "freon". When it leaks out, the A/C stops working, and can put a heavier load on the engine. Perhaps it was over filled? That could make the compressor very hard for the engine to turn.

Has the throttle and return cables lately (like ever) been lubed/oiled?

Also, put some grease on the end of the handlebar where the throttle sleeve rides.

Sometimes on them the throttle shaft which goes in to the carburetor gets rusty and turns hard. If you can't move the throttle butterfly by hand spray pb blaster around the shaft and move it back and forth to unstick the butter fly shaft.

A warped throttle blade could cause this or a binding throttle cable or linkage, you must have to figure out which one.

While I agree that the fuel pump would be the most likely suspect for this car failing to restart, if it is the fuel pump, however, the fact that it started after tapping hard on the throttle would be pure coincidence.
I believe that all e46s have used drive-by-wire throttles. That is - there is no throttle cable between the gas pedal and the throttle. The gas pedal generates an electronic signal that tells the computer where your foot is, and the computer opens the throttle. You say that the car experienced a sudden stop. Did the engine get shut off while the throttle was open? If so, perhaps flooring the throttle caused the electronically controlled throttle to cycle and reset itself (just a theory).
Listen for the hum of the fuel pump when you turn the key to the On position, before you turn the key to the crank position. The hum should last only a second or so. The pump shuts off after the system is pressurized. If you don't hear that hum when the car fails to start (if this happens again), then you have a fuel pump problem.

There is a low speed jet and high speed. Low speed sets the idle or stays running without feathering the throttle./// When setting high speed, hold the throttle wide open until engine revs high, then turn back in(at least ¼ turn) to slow the engine which in turn gives the engine more gas and runs cooler. If you have it too lean on the high speed, it will be hot and hard to start. Sometimes using 40 to 1 mix helps with starting, especially on older chainsaw. good luck

Check the air cleaner and muffler for plugging first. Turn both jet needles CW to stops (lightly), then back out each CCW 1-1/2 turns. Turn the idle speed screw CW about 1/2 turn. This is a basic setting and the engine should start and run. Allow to warm up, then pull the throttle full on and adjust the H needle CW til the engine speeds up somewhat, but is still 4-stroking. Correct setting is when the engine 4-strokes, but immediately 2-strokes when cutting. Release the throttle and allow to idle. Turn the L needle CW until the engine runs fairly well, but still allows the engine to 'follow the throttle'. Adjust idle speed screw so that the chain stops moving, but still allows the engine to keep running. Hope this helps!

it is a dirty throttle plate. $280 is an absolute ripoff for a cleaning, it is about 10-15 minutes of work. They are probably quoting replacing the throttle body which about a $150 part.

Got to Autozone and get "CRC Throttle Body Cleaner". Remove aire inlet hose at throttle body with a screwdriver. Spray a *small* bit onto throttle plate, use clean rag or qtips to remove gum/varnish. Spray pivots and also clean around perimeter of the plate. Take it to an independent mechanic if you are unsure of the procedure.

Sounds like the idle speed motor is not working or very dirty throttle body.
You would have to remove the black plastic hose on top of the throttle body and get a cleaner(throttle body cleaner or carb. cleaner) and spray it down into the opening.
If you have a hard time starting because of the liquid in the throttle body, try holding down the gas pedal while starting.

Your right it is hard. I usually leave the clamp on the hose to the idle vavle and pull it out with the throttle boot atatched. lossening the clamp on the throttle valve isnt easy. I actually have a long flexible driver to do that. you can use a 6 mm open wrench but it takes a few turns on the clamp.