I'm not aware of any mass-market AA or AAA-using device that won't take lithium primaries or NiMH rechargeables.
My Garmin GPSMap 60Cx won't take fresh lithium AAs. The voltage is slightly too high, and it shuts down. If you discharge them a bit, they're OK. It's a pain to have to do that. I basically have a pair of Li AA spares that are more or less dedicated to the GPS.
I use a variety of AAs. Alkalines for cheapness; they get used in low-drain devices like remote controls. Lithium for emergencies and spares, for their long life, lightness and cold-weather performance. Eneloops for day to day use. I have several chargers, including a Maha and ones that will run off a car lighter socket. I also have solar panel chargers that ought to work, although I don't have much faith in them.
One of the advantages of not standardizing is that whatever kind of battery I find, chances are I have a flashlight that will work with it.
I used to prefer diversity too, and I still have some old torches that work on weird batteries, including a Surefire U2. However, technology has now passed it by and I have a tiny Nitecore D10 that uses a single AA and out-performs the U2 in every way (except UI). It's always in my pocket, which the U2 couldn't be for size reasons.
I have found standardising on AA to be hugely beneficial. On my last holiday I took 2 chargers instead of 5. I can move batteries between non-essential equipment, like the shaver or the camera, and put them where-ever I need them most.
The drawback of AA is that in a crisis they will be the first to disappear from stores. You can't rely on being able to buy them at will. Which for me just means I stockpile them in largish numbers.
Since one AA Eneloop will give enough light to read by for 50 hours, you don't really need that many for any reasonable disaster. For very long term disasters, rechargeable is the only way to go. Either solar power, or else siphon petrol from the cars of the people who already got wiped out by the zombies\b\b\b\b\b\bdisaster and use it to power a car charger.