I don’t have an answer - but a similar problem.
I ordered and purchased a new 2015 Expedition Limited 4x4 with the push-button start.
Since new, intermittently the vehicle will display the “no key detected” message after I enter the vehicle with the key in my pocket and start the engine by pushing the button.
The message displays AFTER the vehicle starts. Sometimes, a second message appears (trailer wiring fault) - other times, it is just the no key detected warning.
When new, I took it to my selling Ford dealer twice for the issues. I was told since there was no trouble codes stored, there were no repairs warranted.
Several times I replaced the batteries in both fobs with new ones - with an expiration date years-out. It made no difference.
To this day, I randomly and intermittent receive the warning messages. I have never towed or even connected anything to the receiver, or the two different trailer electrical pin connectors.
I’ve given up on trying to figure out the problem. The vehicle has never not started, but displays the message AFTER the engine starts. I always have the fob in my left front pants’ pocket.
I’ve tried removing the fob when the message appears and moving it around with my hand - very close to the dash and push-button start switch. The engine continues to run and the message remains until I clear it (or shut the engine off).