Thank you everyone. After reading the last message here, I contacted a very helpful parts guy at a local dealership. He said that, after years of use, the regular key gets worn down and will not open the glove box lock although it will open all other locks on the car. This is because, as glove box locks are almost never used, their cylinders remain essentially brand new and still need a new key, while the other lock cylinders have worn down along with the original key.
I therefore am going to chance it and get the key code from the dealer (have to drive there and present proof of ownership) and then pay a locksmith to get a new key made off the code. This requires special equipment and so costs more than the usual copying process. (Incidentally, the parts guy warned against using the new key in any lock but the glove box, as it might cause problems in the worn cylinders. So, I guess I'm gambling a few bucks basically for a one-time use of the key.)
If this works, its better than drilling for sure. And I'll get 5 bucks back when I return the cobalt bit I bought and I won't have to obtain a new lock cylinder.
Moral of the story? NEVER lock that glove box......