Wouldn't such a rumour make people think that if they brought Tesla shares, they could get a share of SpaceX if they merge, given their public stock and the later two are not.

So such reports, in such situations, will be interesting if it has an impact on Tesla shares. Which could trigger an investigation.

SpaceX would only be harmed by the merger. They're a great company on their own, accomplishing amazing things and creating revenue. Tesla is going down the tubes, X still makes some money from ads but is basically just a propaganda outlet, and this new robotics thing is a ketamine dream. Whatever enabled Musk to create SpaceX and Tesla seems to be slipping.