That first link doesn't support your point. No one would claim that all kettles ever sold in the UK have C13 cables. (No one would even claim that none use C15 – after all, some companies will surely just use the same design across all markets if possible.) This particular kettle is before C13 and C15 were even standardised.
The website it's from has a fair number of kettles from the relevant time period (1980s and early 90s). These two (which seem to be variants of the same model) [1,2] have an OKish view of the power connector and look more likely to fit C13 than C15 from what I can make out (no notch). This one [3] is clearly for C15 though, but as I say it's not a surprise that some exist.
On the contrary, it supports exactly the point made in the preceding paragraph, which even pointed out that the IEC standard came later.
The phrase "Should have gone to Specsavers!" comes to mind. All three of your examples clearly have notched connectors. Two have the notches at the top, and the Russell Hobbs one has the notch at the bottom. Their kettle leads were not C13.
So to repeat: When kettle power cords weren't captive, as they are nowadays, they weren't C13. I've already given an example of a kettle preceding the standard that didn't take anything like a C13 connector, and in vainly arguing against that you've ironically produced three more examples of kettles from later decades whose kettle leads were also not C13.
Here's yet another one, where the lead itself is in the picture. It's not C13.
If there had been examples of kettle leads that were C13, I'd have long since used them to really tease my late friend. But kettle leads in the U.K. have never been C13, and my late friend was right that "kettle lead" for a C13 power lead is a misnomer in the U.K..
The website it's from has a fair number of kettles from the relevant time period (1980s and early 90s). These two (which seem to be variants of the same model) [1,2] have an OKish view of the power connector and look more likely to fit C13 than C15 from what I can make out (no notch). This one [3] is clearly for C15 though, but as I say it's not a surprise that some exist.
[1] https://www.modip.ac.uk/artefact/aibdc-001258
[2] https://www.modip.ac.uk/artefact/aibdc-02488
[3] https://www.modip.ac.uk/artefact/aibdc-003345