If you’re assuming “as long as the hardware will function” in the first place: even digital copies, DLC, and updates installed on the system before the servers shutdown will continue working even without hacks. There’s no check-in requirement except for the subscription-locked things like SNES games.

However, the result of a nonrepairable hardware failure when you have no hacks nor official servers is rather bad no matter how your games are obtained: OFW does not allow you to transfer save data from one system to another without going through Nintendo servers and a vast majority of cartridge games are incomplete without updates or DLC.

I never liked to play DS games on 3DS because of the blurry screen: DS games run at a 256x192 resolution while the 3DS screens stretch that out to 320x240. Non integer factor scaling at such low resolutions is incredibly noticeable.

DSi (and XL) similarly can be softmodded with nothing but an SD card, though using a DS Lite instead with a flashcart can enable GBA-Slot features in certain DS games including Pokemon.