Can Elmer's wood glue work on metal?

No, wood glue will not hold a metal tube in wood. It's not strong enough. You will need to use an epoxy. There are five minute epoxy's that work well on metal. I recommend JB Weld, it's a two part epoxy and the strongest that can be used safely on metal. It can be stained, but it will still be "lighter" than the stain on the mahogany.