Improving acceleration using 2bbl Carburetor?

That's not unusual depending on how you prime it (pump thr throttle) and the choke setting. On a '64 factory settings for idle and choke should provide pretty good starting in all conditions. But today's fuels have distilation curves that can make it starting a little more difficult after the fuel has been sitting in the bowl for a while.

I pump the throttle three or four times before I crank it usually, and if it gives me trouble starting I pump it a few more times during. As I’m doing my tune-up I’ll make sure the choke and idle settings are matching what the FSM recommends and see how that does.

Yes the initial timing can make it easier or more difficult to start the car.
I am concerned that the distributor has been replaced and replaced twice.
I'm also concerned about what 'gunked up'.

It was replaced initially when I first bought the car because the previous owner pretty much ignored everything. For instance: He bought a new stereo and speakers and cut the dash installing them, removed the bench seat and installed cheap buckets from a modern Toyota, etc. It’s hard to find any work that he had done that was really for the maintenance of the car.

As such, the starter was replaced with a reman unit and the distributor was replaced, to be safe. After driving it around for a few hundred miles, the car sputtered while driving one day and quit on me. After trying everything I could think of to sort it out, I called a teacher of mine and he came and took a look. He popped the distributor cap off and it was essentially black all around inside. He scraped and cleaned off the points and then I was able to crank it and it ran fine for a few minutes before having issues again. I wish I had pictures. My mechanic replaced it under warranty. That black stuff is what I’m referring to as “gunked”

Well you don't need to understand how they work to make the adjusments. Just follow the procedure.
Doing so may help you pick up on how they work. A lot of us learn like that. We can read about it, and we can work on the parts, but only when we do both are we able to understand what they do.

I figure as much. Adjusting it will help me learn about the process as well as how it works. I’m very interested in it, but trying to wrap my head around it at the moment is giving me trouble. Just form to get out their and get my hands dirty.

Pretty much the same answer I gave before about electronic ignitions in general.
IF the combustion conditions are such that a slightly higher voltage will initiate spark and if more current will build a better flame kernal so there's a good burn instead of a misfire or incomplete burn, THEN yes.
So in other words, if something is going in inside the cylinder that is making it difficult to initiate combustion, then a good HEI or similar setup may be able to overcome those conditions.

What I just don’t know is if it’s an issue with the engine. Given that it has no issue running other than starving for fuel under some idle conditions, I don’t think it’s a spark issue. What I’m seeking is reliability. Its entirely possible that the issues I’ve had related to the distributor are cheap distributors, or the result of road splash given that the distributor sits low to the ground on the slant.

Not sure what you are describing.
A battery that doesn't get recharged will definately make it difficult to start.
Otherwise the alternator isn't involved in starting.

The issue I ran into was that my battery wasn’t being recharged, making it difficult to start. For a few days (maybe 2) i was able to start after a try or two and then it was fine. Just driving 2 miles to and from work. Then the starter would barely turn over. After a jump, it started fine.