I believe the conventional wisdom has it that other issues arise beyond the battery. Car fluids start breaking down, particularly the oils and gasoline, which in turn will start to gum up the fuel pump. Rubber parts start to dry rot. And cables and the like start to rust and freeze up. I know even a 1-2 week period of non-use on the BMW, I can feel the tires have bottom-flattened and the brakes have rusted and seize a bit, and the engine takes longer to start (piston seals harden, oil drained out). Gas over time can also take on water, particularly with the ethanol component these days.
I once left gas in the tank of a mower over the winter. The mess remaining in the spring when I tried to start it again was educational. Should at least use gas stabilizer...
If we are talking about storing a vehicle for well over a year, then yes, other issues can arise. Mostly related to the fuel system and tires possibly developing permanent damage.
As far as the other things you mentioned, rubber parts don't dry rot because of lack of use. They dry rot because of age. By "cables" I assume you mean the parking brake, which most people don't regularly use anyway so again with age or the environment the car is driven/stored in, they will be just as likely to rust and seize up regardless on an older car. Many newer cars have electric parking brakes, so this is a non-issue. As far as brakes go, yes rotor friction surfaces will rust, but if the car is stored in a garage, to get them to the point of rusting enough to completely seize up would take years.
The engine taking longer to start is usually due to some air in the fuel system, not "piston seals hardening" or engine oil draining out. Piston rings, which are the seals I can only assume you are talking about are made of metal, and thus will not "harden" if a vehicle sits. The vast majority of oil that is going to drain into the pan after the engine is stopped does so in the first few minutes after an engine is stopped. Engine oil is specifically designed to coat and cling to engine parts protecting them from corrosion over time and lubricating in the first second or so of startup before the oil pump can supply the whole engine with oil. While an old engine that has been sitting an extremely long time could "seize up", this would take decades to occur.
Gas does degrade over time (if it's less than a year then you really only lose some octane, and pick up some water), and fuel "stabilizers" offer mixed results, especially in fuels that have ethanol added. The likely reason you ran into issues with a lawn mower storing it for a winter is because small engines that are carbureted aren't technically designed for gas with ethanol. Their seals swell and carburetors get gummed up as a result since the have rubber bladders that break down. Cars made in the last 2+ decades do not have these issues since their rubber parts are designed to deal with E10 gasoline and they are fuel injected.
I've been storing cars for the winter (and sometimes extended periods) for years. My current 2003 911 Turbo usually sits from mid November to early April without being moved or started. My "winterization" protocol is to change the oil, inflate the tires to 2 psi above recommended cold tire pressure spec, run the gas down to about 1/4 tank, pour a bottle of HEET in to absorb any moisture, clean the car, close all windows completely, hook up to battery tender, and cover her up. In the spring, I take the cover off. Pull off the battery tender and she starts right up like I drove her the day before. I check the tire pressure and my first trip is straight to the gas station for some fresh 93 octane. Works perfectly every time, and my car doesn't leak a drop of anything.
