From first hand experience, I've seen half of those mountains with my own eyes:
Mount Ngauruhoe, Mount Everest, Mount Fuji, Mount Cook, Mount Rainier, Kilimanjaro, Chimborazo, Ama Dablam, Mount McKinley, and Mount Shasta.
The other half, I have not seen myself:
Piz Bernina, Mount Ararat, Matterhorn, Alpamayo, Mount Robson, Fitz Roy, The Eiger, Aiguille du Dru, Aconcagua, and Shivling.
All those are beautiful mountains, but I doubt the person who chose them has been to many of them and the choice was probably made up in some conference room.
I can't really judge in a fair manner for the mountains I haven't seen, but if I were to choose the most beautiful mountains that I have seen, most would be in the Himalaya (I haven't been to the Karakorum). A few might be in the Caucasus, the Southern Alps (New Zealand) and maybe southeast Alaska, but other than Ama Dablam I wouldn't have chose those ones (even if they are pretty). Mount Cook might rank in the top 30-40 though.
Out of the mountains I haven't seen, the ones in the HImalaya (the ones I haven't seen), Karakorum, Baffin Island and Cordillera Blanca would rank as the ones that look the most beautiful in photographs.