for me the question is, whether the cothes are clean or not and the question is not, whether they look clean.
We (fine bakery ware industry; any kind of coloring raw materials: oil, cacoa, egg) have switched in the hygiene area to white for maintanance too. All cloth is personnalized and can bei identified by bar code. Of cource the cloth of maintanance will be collected and washed separately.
In many cases their is not need to become so dirty if work is organized well and emplyees are educated in this way. Look to Japan. Japanse show us how to manage work not only in food industry. There is no need that people in food indistry looks more dirty than in automotive. If you asked your customers/consumers what their expectation is they will tell you that they expected cleaner people in food. Is this the reality? Asked your lean management team to visit other companies and other industries. I am sure many people in food industry will be surprised.
For improvement you need a long time because you have to change minds and awareness. We have started 10 years ago and we are still on the way. If you reduce contamination/impurification in general the described problem will not disappear, but will reduced to a larger extend.