It was entirely due to 2001.

There just wasn't anyone even close in that time period in terms of visual effects. There were two nominees for Oscar in Visual Effects the year 2001 came out, the other one being Ice Station Zebra. I'll have Roger Ebert describe:

A lot has been claimed for the special effects of "Ice Station Zebra" - which made it disconcerting to find them so bad. An opening shot of a satellite is obviously a small-scale model (and not a tenth as convincing as the space scenes in "2001: A Space Odyssey").

While moon landing conspiracy theories were around from the very start ("staged by Hollywood on a Nevada desert") the actual specific attachment to Kubrick was originally posed in the 1976 book We Never Went to the Moon by Bill Kaysing.

In addition to trotting out all the various confused-science points (like worrying about the lack of stars on the video, which wouldn't have been captured by the aperture being used) he spent a page speculating about "the visual aspect of simulation", noting that the film 2001 was started in the summer of 1965 "at about the time when those who really knew what was happening in the Apollo project began to make their final decisions".

The connect-the-dots here was that since Kubrick did consult with NASA for accuracy, he figured Kubrick could also consult with them and make an entirely new movie alongside his other movie, where 2001 acted, as Kaysing described, an "ingenious cover".

There wasn't much time spent on Kubrick, but Kaysing's book was relatively influential amongst conspiracy-types and the Kubrick theory was essentially accepted after, the most elaborate new developments being thinking that The Shining (1980) contains clues about The Truth (i.e. that the twins represent the Gemini mission, as the characters weren't twins in the original book).