Fauci flat out lied about masks. He knew they were beneficial, but worried there weren't enough, so he lied. And Trump was his boss and allowed that.
When Trump read the belated CDC mask recommendation in a bored monotone he perked up and repeated that it was only VOLUNTARY and immediately went off on a digression on why he wouldn't mask up. (because he has to meet with foreign leaders who actually don't come to the US when it is infected).
So no, Trump didn't say don't wear a mask, but he made it extremely clear he thought it was nonsense and wouldn't do it. Later he opined that many wear masks just to display anti-Trump sentiments, which 1) implies mask wearing isn't necessary and 2) inspired his followers to show Trump support by not wearing masks (as Trump intended). Trump never lets himself be seen with a mask. The WH is reportedly a mask-free zone.
I remember when I first saw Xi JinPing wearing a mask on TV in late January.
The effect was dramatic. It was a potent symbol. I thought: Damn, everybody needs to wear a mask. Trump continually failed that test.
And largely due to Trump's stance, GOP leaders resisted calls for masks until July 1.
Only when the election is getting close and Trump became politically toxic.
Otherwise, the mistake a lot of (GOP) Governors made was following the Trump game plan, including delegating responsibility to counties, not issuing mask guidelines, downplaying the severity, allowing churches to remain infection centers, reopening too soon, etc. Florida the best example of all of that. With the Guv following Trump step by wrong step.