The problems with modeling human disease using "found" cancer cell lines (of which HeLa is just one of hundreds) have been widely-understood for well more than a decade. These cells have absurd, fragmented genomes with bizarre and variable chromosomal copy numbers. They have lived in culture for sometimes many decades, under strange evolutionary pressures. They provide models in which we can explore human-like cells and regulatory pathways, but it is well-acknowledged in the field that these are not sufficient to model a generic form of cancer, much less a particular patient's cancer.
Thiel is approaching biotech with the same blindness with which older investors approached investment in datatech during the dot-com bubble. There is no more reason to believe that this startup will corner the market on xenograft-based research into cancer therapeutics than there was to believe that pets.com would become the be-all and end-all of everything pet related online. But this comes back to your point, that the founders have done a fantastic job of communication. I hope they are communicating the truth, and not just their dream. It will be a big fall if it is the latter.
I really can't tell from the text what kind of cells they use and how random these are, or how novel their method is (if it is even novel).
I was aware of the HeLa contamination issues and ultramarathon-culture length, but I haven't heard the bit about the extreme genomic abnormalities in HeLa (I haven't worked with HeLa cells) and it sounds really interesting, can you speak a bit more about the specifics?