Sorry for the somewhat late replies.
Seeing all these modded Gameboys is making me want one. Just got a couple questions though. Which is better, the original Gameboy or a Pocket? Why not just play Gameboy games on an Advance instead of the original hardware?
No original (non-modded) model of the Game Boy is perfect. All of them either have some kind of crucial flaw, or lots of tiny ones.
Still, most enthusiasts (myself included) use a GBA SP 101 to play regular Game Boy games, because it's more convenient and overall is the least flawed of the bunch. If not then get a Game Boy Color, as that one is pretty much perfect outside the lack of GBA support. The other simple alternative is emulating games in whatever device, which is doable due to the original Game Boys not being that nice to begin with and GB emulation is very well done.
From what I understand (as I have yet to get a modded Game Boy), the best modded one would be a regular GBA with a backlit screen. Even then, merely replacing the regular GBA screen with one from the SP 101 still wouldn't be optimal, as that one is a bit blurry, so it defeats the purpose of trying to get the best possible using original hardware. That means the backlight mod has to be done to the regular screen, and for that you need to contact someone that's very, very good at it. I remember there is/was an Australian user that did that job very well (maybe I shouldn't have backed out from getting one from him).
Does the EZFlash shorten battery life compared to playing normal games?
I want to play Mother 3! My preference would be to get one of the old-style GBA flash carts that don't use SD cards, but they seem to have disappeared off the face of the earth.
Yes, but not by that much. What happens is that the LED shows there's low battery, but in actuality there's hours left of usage, and sometimes it even goes from red to green.
If the EZFlash IV can run Mother 3, and you have decided to use the original hardware, then go for it. I know it uses micro SD cards, but copying single games is very simple.
Someone recommended you an EverDrive. I wouldn't get one in your case, because EverDrives are quite expensive and Krikz takes many years until perfecting them, that via upcoming separate hardware releases. If we go by what happened with the N64, then the 2022 ED GBA v3.5 will be perfect, after updating to the newest firmware released afterwards. Instead, the EZFlash IV just works, as it has for over a decade, and has 99% compatibility (not with the Famicom games and some Dragon Ball ones I believe).