I've always just left it attached to the main part. My family's always sliced the pineapple into disks then cut those into 4ths. You end up with a small part of core per piece.
If you want to get more out of pineapples did you can use the rinds? It's a fermented drink called tepache. It's the rinds of a pineapple, water, and a type of brown sugar called piloncillo. I've never heard of using the core in it but I dont see why you couldn't add it in. I say add because the rind is where the fermenting bacteria comes from.
Ha I also just finished watching the first season. Based on the time you posted this there's a good chance we were watching it at the same time.