Maybe it's just me, but I'd be wary of coaching at a mountain that wouldn't allow me to freeski in their uniform, especially if they force me to wear it when on the job. What do you do when there's no one on the slopes at diner time and you have 1 or 2 runs to yourself before the afternoon session begins ? You change clothes ???? Plus, it corroborates the vision that many have about coahes and instructors that "those who cannot do, teach".
It also seggregates the instructors in two groups: those who are good, and those who are so bad that the mountain would be ashamed of the public seeing them doing their thing on the slopes. That sucks big time for team spirit. Many of the coaches I've worked with were incredible skiers, but many were also barely meeting freeskiing qualifications, and many were as good at coaching and developping racers that their "abled" peers.
And for those who see it as tacky or as gaperrific, I'm very proud of letting my skiing do the talking. One of my guilty pleasure is wearing the most fluorescent garb, growing a mullet for ski holidays, skiing old skis, etc, so I'm actually having fun with the whole gaper thing. Fashion ain't my thing
Finally, who needs to spend money that would be better off on gas and equipment when you have a perfectly serviceable coat sitting at home ? If the resort doesn't allow it, so be it, but if there's no problem, I wouldn't see why you would have to buy a jacket just because you're afraid of what other people are thinking of you.