Fact is - it's unprofessional. You wouldn't go to a restaurant to see people chewing gum. Same for airline staff on the ground, and indeed in the air.
I've got no problem with crew chewing gum, eating, etc. out of direct passenger view. However...doing it on the cart is verboten.
WHEN YOU'RE DOING YOUR JOB IN UNIFORM, YOU HAVE AN IMAGE TO UPHOLD. Can't deal with it? Leave. Someone else will be able to 'hack' the inability of chewing gum in front of passengers for another time.
It's called a service industry. Nothing new under the sun. Can't figure out why people in this thread are so shocked by the notion that you SHOULDN'T chew gum in front of pax!?