It's a bit of an apples and oranges thing... First the lithium ion batteries: If you have 2500 mAh and you pull 2.5A, you are going to run dry in one hour. And if you do the same with a 600mAh battery (14500), you'll be lucky to get 15 minutes. Then the alkalines: first of all, you might need 3 of them depending on what kind of voltage you need. Second, a heavy duty AA alkaline will give you about 1500mAh. However, if you run three in series to power up something like an Arduino, you get triple the internal resistance and you still get 1500mAh (albeit at 4.5V). I don't know how long the AA alkalines would last though, but it will be far less than 1500Ah/2.5A (36 minutes), because the battery rating is based on a constant 50mA current draw and 2.5A is 50 times that much.
It's quite possible that you can get a higher max current from a larger cell...likely even. In fact the second link that I posted says that a D size alkaline has a 0.1Ω internal resistance and an AA size is 0.15Ω, so that means the larger alkaline will have a lower voltage drop than the smaller one when you use it in a high current application.
It's not 100% clear to me if Dio is having a problem because of maximum current draw when the servo is being used or because the battery goes flat too quickly. These are two different things. Cars (the gas-guzzling type) still use lead acid batteries because they are excellent for short term burst power (starting an engine). Apple's first Mac Portable had a lead acid battery, but laptops have moved on because they generally need a relatively flat amount of power for as long as possible.
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