Would this be possible/make sense? I was thinking of adding an extra jack port to one of my guitars, for some experiments with different amping and stuff. Would this work, or would it just sound horrible or break the thing?
This may be a bit misleading, so I'll clarify... you aren't going to hear the effects connected from one jack on the output of the other jack. The way that your signal chain is going to impact the sound is by loading. If you have 2 output jacks driven only by the pickups, you are going to end up with weak-sounding signals with both jacks connected to amp/signal chain.
But yes, you definitely want to either buffer the output or just use an ABY box to split it. I think the ABY box will be much simpler. If you're into DIY, you can even make your own. Just 1 link to get you started: