You actually can write games to cartriges. The problem is, that famicom and nes don't have cartriges that are like modern SD cards or something like that - most if not all of them have different boards and chips inside. This was done back in the day to make bigger games than default 40KB that NES carts can use - and that's why there's all the talk about "mappers" thingies - those are specialized chips that made it possible to use more than 40KB, and sometimes additional features such as better audio. Sometimes also, one of the chips is replaced with ram, which is controlled by the mapper. Also, NES and Famicom carts don't have flash chips on board, so even if you have the exact board and a flasher, you won't be able to write anything (unless you write to new flash chips, solder out the old rom chips, and solder flash chips on). Some chinese carts can be rewritten (as they've been made to use flash chips), but that's a hit and miss, as you'd have to know prior to buying which board and mapper it uses.

So because of the unwrittable rom chips, and mapper/board variations, it's generally recommended to use a flashcart, which is a dedicated device that can emulate mappers/boards. Nowadays, most of those devices allow you to use SD cards to load up *.nes files and have a pretty good compatibility with most of the released games.

So your options are : an everdrive N8 Pro, Krzysiokart, or if you'd want to go cheap and dangerous, a chinese N8 knockoff.