Color changes in ground beef?
You are correct in saying using color to determine doneness is less reliable in older meat. As the meat is exposed to air, it oxidizes giving it a brownish color. When checking for doneness, people sometimes see this oxidized brown color and mistaken it for being fully cooked through. However, this can occur for fresher meats also.

According to the USDA using color is not recommended to determine the doneness of hamburgers. 


  Some ground beef may appear to have lost all pink color before it is fully cooked. If raw ground beef is somewhat brown already, it may look fully cooked before it reaches a safe temperature.


In some cases, meat that is already fully cooked can have a persistent pink coloring. This is usually more common in leaner meat.


  Some lean ground beef may remain pink at temperatures well above the 160 °F final cooking temperature recommended for consumers.


So it is really recommended that you get a meat thermometer to test the doneness of your burgers in the future.

With all that said, unless you have really thick giant burger patties(10oz +), cooking the burger at 300F for 15-20 minutes should definitely be enough time to insure that the burgers are fully cooked through.

Reference: USDA on Color of Cooked Ground Beef as It Relates to Doneness