My battery still works in my Link's Awakening. The batteries in these Nintendo games last a long time given their predicted lifespan(I think 6 or 8 years?). However I wouldn't start a new file on my Link's Awakening as the battery could just spontaneously die in the middle of my playthrough. Safer to play it on a newer medium, like 3DS or rom or something.
True, my Pokemon Gold is like 12 years old. Pretty sure you can replace the battery with the right tools, but I think it also deletes any save you currently have by doing that. Heck of a lot of people advertising GBC cartridge repair on ebay.