This is an example of a media trope in headlines that I've long felt deserves a specific term.
It's where the headline writer links two current issues which IRL may not be cause and effect. They're often linked with "while", this time it's "as". Coincidence is not proof of causality.
In this case, Starmer's ratings have been tanking throughout this "honeymoon period" of his government (and not without good and easily identifiable causes). I doubt any goings-on involving Musk have had a significant effect, and if they will, it's going to be a while before they percolate through to be detected by any reliable polling.
For those who were willing to listen (and I'm certainly no fan of Starmer), he was able to give a firm rebuttal of Musk's tappety-tap rabid ravings because his record as Director of Public Prosecutions was pretty good on these issues, none of which are new in the UK because we've chewed over and agonized about them publicly at great length for more than a decade. Indeed, the Tommy Robinson video that made such an impression on Musk that he became an instant expert on the subject matter and UK politics as a whole has been doing the rounds for a while, has been roundly debunked, and Robinson's currently in jail because of some of the lies it contains.
The UK had a full inquiry report in 2022, and one of Starmer's most telling lines was that the time for inquiries was over and it's now time for action. He was also able to point to the fact that Jess Phillips (again, I'm no fan) has done far more to safeguard women and children during her political career than Musk ever has or will, or Tommy Robinson or Farage for that matter. Phillips's call for local inquiries rather than another full UK one is a coherent stance in this light, and one she's been repeatedly libelled and threatened for by vicious goons and goonbots stirred up by Musk's nonsense. Starmer even managed to show a hint of fire during his speech, which is a first.
That won't stop the UK Tories and Musk and Robinson et al. fanboys whooping it up, of course, and I've no confidence that Labour can keep up a consistent and persuasive public line on this faced with the deluge of misinformation. If they do, it'll be a first in this term of office.