Updated 2021-06-04: The k-means++ implementation I was using previously appears to have been flawed. I've updated results using a better implementation.
The above report focusses on R. @ctwardy has replicated the basic result here and done some further exploration in Python.
Updated 2021-06-19: Added Appendix 2, sketching an argument that the asymptotic density of k-means++ is optimal.