The ZIF connector on the original NES was unreliable. The carts are perfectly reliable. The NES 2 does not have the ZIF connector and doesn't have issues.
Depends on the cartridge. Any of the ones with a battery save (like Zelda or Final Fantasy) need to be opened up, have the welded in battery clipped off the board, have a CR2032 socket soldered on instead, and then put a new battery in and reassemble.
It's a LOT of pain to get a NES cart with a dead battery working again. That's probably why they opted not to put a slot in this thing -- many people's dusty old carts are dead, and not knowing any better, they'd blame the new console.
Replacing the battery takes a couple minutes. Many of those carts are still running. The number of decent NES games that used battery saving can be counted on your hands, so to call it a LOT of pain might be exaggerating a little.
And while overall most NES games didn't use a battery, many of the most popular ones (many of the ones included in this thing, in fact) do need a working battery. Final Fantasy, both zeldas, startropics, tecmo bowl, crystallis, kirby's adventure...
Crystalis absolutely has a save/continue system. You can see some guy demonstrating the save system in crystalis in this video, complete with save functionality: https://m.youtube.com/watch?v=73za3LHWzwk
Incredible. I owned the game for years, played it a lot, beat it once, and never noticed a way to save, so I assumed you couldn't. I even had the manual!