While I do not dispute that there have been no obvious links found between wild deer performance and mineral supplements, I cannot ignore the undeniable use a mineral site gets while a doe is pregnant/nursing or a buck is growing antlers. Perhaps the difference is immeasurable on our scale or possibly there's a metric we are not currently considering. I don't know. But something is happening in a deer's body while it's in a state of accelerated growth that makes it crave minerals. If they were getting all they could use naturally then they wouldn't be fighting over a mineral lick. I've seen does turn Mike Tyson on other deer to keep them away. I don't see that kind of territorial aggression often or in many other situations but it's common at mineral sites. To my layman mind that means they are getting some kind of benefit to their growth needs even if science hasn't yet detailed it. Heck, it could be something as simple as alleviating a tummy ache due to eating poison ivy. Who knows? Regardless, it's apparent something is going on.
I once saw a documentary on south American parrots eating mud and the proposed hypotheses were quite interesting and potentially parallel what could be going on with deer. One notion was that the clay/mud helped balance out the overly acidic and toxic plants the birds were eating. Another was that it supplemented their diet with minerals. Here's a quick read article about it. The similarity between how the birds use clay outcrops and deer use mineral sites is uncanny.
https://www.npr.org/sections/thesalt/2017/09/07/547981850/why-do-parrots-and-people-eat-clay