Personally I would definitely recommend proper flash carts with SD card in it, mostly because the other kind isn't really produced any more and compatibility is spotty, or if you do get the proper made ones you will pay nearly as much as what the more fancy EZ-Flash or Everdrives would cost.
For example, insideGadgets makes really good ones:
- GB, 4MB, 128kb FRAM Flash, supports a few mappers:
https://shop.insidegadgets.com/product/gameboy-4mb-128kb-fram-flash-cart-low-power/ - $43
- GBA, 32MB, FRAM Save:
https://shop.insidegadgets.com/product/gameboy-4mb-128kb-fram-flash-cart-low-power/ - $35
To actually flash the games you will also need a flasher:
https://shop.insidegadgets.com/product/gbxcart-rw/ - $30
All prices without shipping or import fees (they are made and ship from Australia).
BennVenn also makes similar things, and prices are similar as well.
Downsides to those things: Compatibility is nowhere near as good as either any of the EverDrives or EZ-Flash carts, you have to keep track of more things (flasher + carts), you can only have one game on it unless you modify the roms.
Advantages over ED / EZ-Flash: Much better battery life, true plug and play once they're flashed, no menus or similar to deal with.
For romhacks in particular I would recommend either a cheap EverDrive (GB X3 for GameBoy, GBA doesn't have a "cheap" one) or the EZ-Flash Jr and Omega. Their much better compatibility and ability to handle all types of save formats means you can easily get them to work even if the programmer did something funky with their hacks, and you can just update the romhacks by plugging the SD card into your computer and dragging the files over.
If you have to pick one over the other, I would suggest going with an EverDrive GBA Mini or EZ-Flash Omega DE, and between the two whichever is either cheaper for you in particular or has the features you care about more. The ED GBA Mini takes a bit less power to run than the Omega DE and thus will allow your battery to last longer, if that's relevant to you.
The Goomba Emulator is also preinstalled on both the ED GBA Mini and the EZ Omega DE, the GB/C games are all drag and drop for those two as well and you don't have to worry about anything. Only downside is some games sound weird and compatibility with GBC games is spotty, for example Link's Awakening the water fall sound is atrocious via Goomba Color and Pokemon Crystal just will not work correctly at all. The ED GB or EZ-Flash Jr will not have those issues at all.
EDIT: I am using some of the insideGadgets flash carts, and particularly the flasher to backup my saves on real carts, but I would only recommend them if you know for a fact they are a good fit for your needs. Support is above and beyond any other company I've dealt with before so I can highly recommend them, but again only if you know for a fact the products do what you want them to. If you have questions Alex, the owner, has an open Discord server and he and other people there are very helpful when it comes to questions about the products or if you've found a new a problem.