I truly prefer when a game uses a battery save (I am really talking about cartridges; for consoles that use some form of a memory card, this is not what I am talking about), but there are some major exceptions. I am going to talk about the Famicom first since this is the one I can get done with first.
I do not own a Famicom. I know that the cartridges come in all sorts of configurations. I also read that to get inside the cartridge is very difficult. Therefore, I think I would prefer a password save instead of a battery save. I really would not want to even try to get inside a Famicom cartridge. I think that subconsciously this is one reason why I have not yet bought a Famicom. (I have now.) The Famicom Disk System, though, is a completely different thing. I would want it to have a save system instead of a password.
I also want to avoid any Japanese Mega Drive cartridges that use a battery save. I read that the screws to open the cartridge are behind the little back warning sticker. Now obviously this is not as big a deal as the Famicom, but I would still prefer to not damage the sticker. The Genesis is not a big deal to get into, just like the majority of cartridges.
Let me also talk a bit about the GameGear. I replaced a battery in a game, but I broke the little tabs that secure the case together. If it was not for how these cases were designed, I would include the GameGear in the "I want a battery" pile. But I am not so sure. Now, I don't really know how to solder, but I still managed with this one game. I think someone who is better would not have this issue. But anyway, there is one screw in the back of the GameGear cartridge, but there are also I think 4 tabs also holding in the cartridge. Honestly, these tabs feel strong enough to hold the cartridge together without the screw. But it's these tabs I don't like.
Now what about the long term? For most (reasonable) cartridge assemblies, yes the batteries will need to be replaced, but it's not that big of a deal. For Famicom and Japanese Mega Drive games, though, it will be more difficult. So I think that password saves will probably be preferred since there is no battery to worry about. But this is one advantage of password saves on all systems. There is no battery to worry about. But I still prefer battery saves on most systems.