Humans are really, really bad at safety science
It's very strange. Humans are actually terrible at assessing safety risk. They drive carefree to the airport then are terrified during cruise, which is really backwards! We see people get ulcers over well-contained asbestos hardboard on their homes... not at all because of any medical understanding of asbestos, but only because late-night TV is plastered with ads from lawyers hoping to get 1/3 contingency fee for tapping a few asbestos trust funds (which benefit career workers who breathed asbestos dust for 20 years without PPE). And SMH - aluminum wire. Which turned out to be institutionalized incorrect installation, easily corrected in hindsight. (and footnote: the necessary torque screwdrivers finally required in NEC 2014, every electrician fought those the way you fight boots).
In simplest terms, people are not inherently good at science. Science is not easy, and the hardest part is the scientific method - squishing out anything that would tilt the results toward an outcome.
Without science, you can't do accurate risk assessment.
But humans are really good at confirmation bias
And you really do not like those boots.
So naturally, you are cherry-picking evidence that says the boots are bad, and willfully ignoring the body of evidence that says the boots are important. If you're looking at all.
You think yourself the grand master of safety experts - don't we all! And you fall into the familiar pattern of bristling at some government requirements that you imagine comes from know-nothing bureaucrats. But that's a fantasy.
Reality is you are quite new to the field. And that sophomoric feeling - "I know a little, therefore I know a lot" actually has been explored by two scientists named Dunning & Kruger. They determined the bottom cohort in a skill area were convinced they were much smarter than they were. They resisted acknowledging their poor scores. Amazing paper. The paper showed training would work if the test subject agreed, but training is only possible if you want to learn, and "confirmation bias" is the definition of not wanting it). I interpret as "It takes constant vigilance to NOT be that closed-minded person".
Especially when you're new in a field.
Managers have given up talking sense into workers.
That's why the sign says "No boots, No work" and not "For why we require boots, see these 11 scientific papers". Because most workers don't give a damn what those scientific papers say. So all workers get treated like 12-year-olds.
Complain to someone about this like my manager or the safety coordinator. I really don't want to do this either since I'm new and I don't think it would do any good. I think they'd just end up pointing to the policy and say that I gotta follow the rules.
That's correct. They're NOT going to have the scientific discussion with you, because they don't think you care. Are they wrong? Does a scientific paper or data-driven study possibly exist which will turn you into a boot evangelist?
Put on your "research scientist" cap and start doing a meta-study of the literature. See what the data tells you. If you can earnestly see an opening for "boots hurt agility" argument, then start developing data and write a scientific paper of your own. Submit it for peer review and see if it gets torn apart by experts. Are you likely to sway? If you admit it is not likely, then you are admitting the established science is probably correct.
I don't like wearing seat belts, but there is no question in my mind the science supports their use. I could sit there and pretend the science does not, but that would conflict with other values I hold.
Generally, moves which require agility should not be done
In your craft, I think you'll find that they don't really want you doing things which require agility to do successfully. Think about a 65 year old obese worker who's sedentary other than this job, and close to retirement. Should that guy be doing that move? Then neither should you.
Any competent employer in your craft will favor safety over speed - because they've tried the other thing, or seen the fate of others who tried the other thing. If you are in a safety conscious shop, count your lucky stars - because the other kind is awful, and can and will get you maimed.
Everybody goes "If I die, I won't care then" but they don't think about being maimed.