wanderover,
You're not wrong.
A few years ago, I was asked to show a party of visitors around a NATS unit.
At the 'de-brief' afterwards, a lady said she had enjoyed the visit immensely, she was slightly less apprehensive about flying, as a result, and commented on the "scruffiest collection of highly-trained professionals" she had ever seen.
O.K. it was a Saturday morning duty.
A year back, on the LAG I was working on, only myself and the 3 lady ATCOs/ATSAs had 'shoes' on. Everyone else was wearing 'trainers'. One ATCO was even wearing 'builder's trousers' too, with big pockets for hammers and chisels etc.
P.S.
I used to wear my slippers on TWR night-shifts though. Nice'n cozy!