Should lemongrass be edible or just a flavoring? In my limited experience with lemongrass (which I buy fresh-ish in plastic bags from the fridge of my local asian supermarket - not frozen or dried or anything), it's hard and impossible to chew, no matter how long you cook it for. I usually smash it, split it down the middle and add it to soups or other dishes with a lot of liquid, then take it out at the end like a bay-leaf.

However, I've recently seen recipes in which finely chopped lemongrass is put straight in a stir-fry or a sauce, which just seems like a good way to end up picking bits out of your mouth to me. Is there a problem with the lemongrass I'm buying? Is it that only the green is hard and inedible? Or am I cooking it wrong?
If you actually chop it finely, you should be okay. Specifically, you should cut it into thin disks against the grain first, so that you're cutting the fibers into short enough lengths not to bother you. Depending on how tough your lemongrass is, you may have to remove some outer layers to do this.

At that point, it may already be possible to chew, but further chopping should make sure of it. If the pieces are small enough, there isn't really anything to chew, and it's not too likely to stick in your mouth given that it's mixed in with plenty of other stuff in the dish.

That said, there's some variation in the toughness, and it does take a reasonably sharp knife to do this if the lemongrass is on the tough side, to the point that it might not be possible. So if you have issues chopping it finely (or too little of the stalk is tender enough to do so), falling back to your usual approach of infusing it into things is the way to go.