For those who have tried, which one is a better option:
Obviously $75 is more than $13 + $5 for batteries, but they wear out and the train's speed starts to slow down, before you want to change the batteries. If I spend $75, what other benefit would I get other than not buying batteries?
The main benefit is consistent performance. The rechargeable battery puts out almost the same voltage until it's almost flat, and whatever current is required with little voltage drop. Disposable batteries have more voltage, but cheap ones can't supply much current. Expensive ones can supply the current, but cost more. You'd be lucky to get 5 sets of those for the cost of a rechargeable battery.
Note that the IR controllers / receivers limit the output voltage to 9V and the current to what the motors can cope with,so there's a limit to how much performance you can get out of the system. But disposable batteries will sit at 9V at least initially, where the rechargeable battery will drop from nearly 8V to 6V as it discharges. If you buy expensive lithium single-use AAA batteries they will cope with the high currents demanded by the LEGO motors for longer, so you will get better performance than any other option, but at considerable cost.
This thread on Eurobricks has a lot of detail, and there's more here
The third option is to build your own battery packs. Either just by plugging "random" batteries into the IR receiver, or by modifying a LEGO battery pack to take different cells. It's apparently fine to put three lithium rechargeable cells in series into one, or 8-9 NiMH ones. If you start with a 6AA battery box it's easy to fit 8 AAA NiMH cells in, or either box can be modified to accept three LiIon cells (you need the right size cells, obviously). Some of the Technic builders are just making packs with three 18650 cells and not worrying about making it look like LEGO, but there are quite a few who build extra cells into official battery packs.
Obviously, if you want to use the train a lot, you will want to use a rechargeable option. You didn't explicitely address this in your question, but even with AAA batteries, you can either use disposable batteries or rechargeable ones.
The cost of using disposable batteries can very fast become prohibitive, but rechargeable ones tend to cost more upfront (especially if you need a charger as well). Even there, you have several options, including "stay-charged" batteries which you may want to consider if you need the train to be available as soon as you want it instead of having to recharge the batteries prior to using it, which can take a long time especially if you don't have enough chargers for all batteries at the same time.
So the first thing is to consider if you're already using rechargeable batteries for other things - if you already have a good charger or two and enough AAA batteries, you'll of course want to start with that and judge if it's enough for you. If you don't have any chargers or batteries, then I'd recommend to start with the official LEGO battery right away, as the cost difference won't be that high. Of course, if you're somewhere in between (say you have chargers but no AAA batteries), you'll have to ponder which route is better.
Another factor which may influence you is the convenience. Using AAA batteries and chargers can be cumbersome (again, especially if you've not enough chargers to charge all 6 batteries at the same time), and even the action of swapping the batteries in the battery box can be annoying. The LEGO battery doesn't need to be removed, usually just removing the train roof is enough.
All in all, I prefer the official LEGO battery; note that you could spare some money on the charger if you manage to find one which is compatible (although it's officially not recommended).
As for the battery box itself, either you buy a motorized train and you get one with it; either you want to motorize another train and then it's probably worth it to buy the basic city train anyway to get all electrical parts rather than buy them separately (since you get motor, receiver, battery box, remote control, and tracks, it's usually a better deal even without considering the rest of the parts).
