You're absolutely right. I've never understood the mindset nor the desire for Ammo Techs to wear that badge. I know of some guys who have done the course were told to wear it. Shows the influence of the RE at the school
I knew an RAF Medic who wore Canadian wings above his DPM jackets right top pocket (around 25 years back) because he claimed to have done one parachute jump while on exercise in Canada. He'd never done any parachute training in the British Forces that I knew of, and certainly didn't wear a British parachute badge.
Parachuting and balcony work just seems to bring out the Walt in people.
Not as seriously as a when one of the Brits does an ear piercing whistle on the run-in for a Bn size night drop into Hunter Ligget training area. They tend to get all serious, close the doors and do a mini interrogation on all the chalk.
Oh and never jam a finger up the loadie's arse as he's using your shoulders to walk on during low-level. It's classed as sexual assault apparently.