You can disregard aluminum as associated with the disease. There was supposed to be a old study that found higher levels in Alzheimers patients, but later studies found no difference. You get aluminum from other sources (tea, etc.) but none of the other sources show any correlation with the disease. Some kidney patients get high does of aluminum in their meds for years, but they aren't more prone to Alzheimers. The direction that the most hopeful studies are taking in seeking a cause do not suggest mechanisms where aluminum involvement seems likely.
Aluminum cookware doesn't hold heat especially well. I could hold heat well, but it would have to be very thick to have a sufficient mass. It does, however, conduct heat very well, and the two virtues of that are rapid heating and less tendency to hot spots. But it also reacts with a number of food substances. As for stainless steel, classics like Revereware coated the bottoms with copper to take up and diffuse heat better. Otherwise, plain, thin, stainless had bad hot spots. But it only works to a limited degree, although they do heat very quickly.
The happy middle ground is stainless with a heavy ply bottom. The heavy bottom contains a layer of aluminum and/or copper. You get the non-reactive stainless steel and the layer of aluminum or copper protected from the food but able to diffuse heat well across the bottom. Some makers offer versions with the ply all the way up the sides at added cost. I ask, though, how much cooking do you do up the side of the pan?
The other very good cookwares are copper, which is VERY expensive (but beautiful). It is, for most purposes, lined with tin or steel to prevent reactions, and if it's tin, it will one day have to be sent off to be recoated. Cast iron, such as is made by Lodge, is fabulous, but it takes more care and has something of a learning curve to not have it require too much maintenance. (You can easily clean up your stainless steel disasters. Iron disasters are harder to deal with.) And ceramic coated cast iron is also nice. All are "non-stick" when used properly. But without taking a survey, I'd guess that most people here do most of their cooking on tri-ply stainless and have one or more of one or more of the others for particular uses. It does NOT have to be the most expensive to work well and have all the good qualities of tri-ply. (But buy them with steel handles, so you can also put them in the oven.)
I use a set of Tramontina stainless, but I have some copper picked up at the flea market, one Teflon type non-stick pan that's only for omelets, a cast iron chicken fryer, and a ceramic coated cast iron dutch oven.