This goes back to portability and, to be honest, you may want to take your tray with you on a vacation or other such event. Heck, you may even be going to visit a friend and want to use it for dinners. No matter the reason, most people are aware that airline restrictions can be strict. It all comes down to safety so it is understandable, but it makes preparing for trips much more strenuous.
Much like the previous question and the FDA, your answer to this and any other airline questions can be answered by the Transportation Security Administration (TSA). Based on what the TSA mandates, bringing a defrosting tray on a plane should not pose any threat or concerns. It uses no electrical components and is not considered to be a sharp object. While you should still check with your local airlines to be safe, it should not be much of an issue.