Is the discharge rate for all AAA batteries the same?
I asked myself much the same question a few years ago. My local very cheap supermarket had an offer on some very expensive name brand AA batteries 'U', so I bought a pack of those, and a pack of their own brand alkaline AAs 'W' at 13% of the name brand price. I also had some own brand cells from a middle-price supermarket 'A', and some rechargable NiMH cells 'N' for comparison. Here is the discharge graph.

As you can see, the most expensive cells are the best, but not 8 times better. The early drop in voltage on the cheapest cells suggest a higher internal resistance, which means they won't be able to supply the power output of the better cells.
You asked about AAA and I'm presenting results on AA size. They are the same technology and build method. The main difference is that AAs have a higher capacity and power output than AAAs, and have a better ratio of output to weight, size and cost than do AAAs, as their ratio of active chemical to case is better.
To get the best delivered energy from alkaline technology, you will need to work from a wide voltage range. A switching regulator is nearly essential here. Once you have that, then a cheaper cell will simply give you a shorter lifetime.
What you do get with a name brand cell is some sort of assurance that the performance for future purchases of the same type will continue to be as per their datasheet. I was unable to find a datasheet for either of the own brand cells.
Note that this is a sample of one of each type of battery, measured once, at one load resistance. I would not be surprised to see the results rank in the same order at other load resistances, but have not done the measurement.
Shelf life is a rather different parameter, and impossible to measure in one day as I did the discharge curves. It depends on the chemical system chosen for the cell (the same in all cases) and the quality with which that has been implemented (as can be seen from the graph, there is a demonstrated difference in capacity and internal resistance). It's perhaps not unreasonable to hope that there might be better purity of the chemicals, or additional additives to suppress leakage, in the name brand cells. However, without specifications or measurements, that would be just hope.