Before I restart Pokemon Emerald, is it worth it to replace the battery?

It's not crucial to the game or anything, without the clock your berries won't grow (so planting them is a waste) and a few other daily events (lottery, berry hand-outs) will only happen once. Actually speaking of Feebas I think this makes it kind of hard to evolve. You won't be able to go to Mirage Island either. Still almost everything in the game works, I've replayed Gen 3 a couple times without the battery and it didn't change the experience very much.