Does a battery's ability to supply high current mainly depend on its internal resistance?
Short answer, yes you are right. The Ri of a battery limits the current it can supply, but the Ri is not the real cause, more a symptom. The design and characteristics of the electrodes, chemical processes, temperature, etc. all kinds of internal and external parameters interact when current is "requested" and Ri is just your way to put all these influences into one handy value. And we need to mention: Ri changes while the battery operates.
For example, my 50km/h RC car does not move a mm when the LiPos are 0°C or below. Temperature is a huge influence on Ri. In the summer 2 of these packs should start my car easily.
Well, there's no little resistor inside the battery – instead, "internal resistance" is our model for the proportional-to-current draw voltage drop you'll see with any real-world power source in some current region.
So, as schnedan said (go and upvote their answer!), "internal resistance" is the symptom, not the cause. Of course, actual ohmic resistances in conductors and electrodes contribute to that – but so do limited reaction speeds and limited speeds of ion transfer.
It is the other way round.
For the sake of simplicity, we model the battery ability to deliver high power with something called "internal resistance".
The model is pretty much basic and in particular really bad for high currents.
The "internal resistance" of an electrochemical cell is non-linear.
It consist of more-or-less ohmic behavior of the electrolyte, the exponential V/A behavior of the chemical reaction itself and has a high current limit because of the limited diffusion of the substances to the reaction zone. And don't even get me started on AC-related (transient) behavior of a battery.
All of these also depend in complex way on the temperature, state of charge and the past discharge history of the cell.
That's why, for example, car batteries' ability do deliver power is rated in CCA (cold cranking amperes) and not by their internal resistance.
... and AA cells are not the best bet when trying to start a car, unless the car is manual, you are strong enough to push it (or have a helping hand or a favorable slope) and you know what you are doing.
A typical car needs ~150A @ 9V in order to start reliably (the battery is 12V when idle and sags 2-3V when starting)
There are ~3Ah, 30C or 50C rated LiFePO4 cells that are pretty much able to do so when used in series of 4, without being much more in volume or mass compared to 8 AA cells.