Do the Trumpsters care that Trump never has proof to back up all his wild claims?
No, because the same way those people come to conclusions is the same way Trump does; it revolves around the same dynamic as those who think "he tells it like it is".
Their opinions come from them stumbling across some click-bait blog post or quasi-news piece that is either totally false or so slanted as to be effectively such, followed by them saying something to the effect of "I knew it all along" (even when they didn't). They then begin to yell it from the rooftops without any evidence whatsoever. Sound familiar?
They love him
for it - not i
n spite of it - because it's the same way their own feeble minds work. It's the natural progression of the TV Dinner generation, being raised to believe everything they see on a screen, as well as their sad little offspring who didn't manage to escape its clutches and inherited the condition wholesale because "muh traditions" have resulted in them shutting out any beneficial strengthening that an advanced society can provide with respect to critical thinking/worldview, leaving only that which existed decades ago rolling around in their heads.
As much as they like to cry about "fake news", they are by far the most susceptible to believing anything that flashes across a screen with pretty colors and crisp fonts.