among larger, alliance-affiliated airlines I know the following do:
All the LH group airlines (LH, Swiss, Austrian, Brussels), both Korean carriers (Korean and Asiana), Iberia, Royal Air Maroc, ITA, Turkish, LOT, Ethiopian, Air Canada, Aeromexico, Aerolineas Argentinas, and (well..) Aeroflot.
Etihad does as well, but at a price: they require purchasing an extra seat for the pet plus a $1,500 pet fee. Thatīd be a better value proposition if they would not impose the industry-standard 8kg weight limit which they unfortunately do.
And then there is specialty airline La Compagnie, a French all-business-class airline flying between New York and select European destinations, mainly Paris. They allow pets in the cabin with a 15kg weight limit, which is the most generous I know among regular scheduled passenger service. This is not a coincidence - when they first started out they did not allow pets in the cabin, but obviously they saw a business case for becoming pet-friendly around the same time Air France stopped to accept pets in business class. Thatīs been lowered to 8kg starting March 2025.
ITA as the most generous of the alliance-affiliated airlines has a combined weight allowance of 10 kgs. Aeromexico has a combined weight limit of 9 kgs. All others are at 8kgs, the Koreans allow only 7 kgs.
Aeroflot carries pets free of charge on 'maximum' (=full fare) tickets. This is promoted as official benefit. I think that's unique.

United does not allow pets in business class on longhaul flights (Polaris seats). That video must either have been older, or the dog was not a pet but an ESAN/SVAN (now much more restricted as well, or the lady in question some kind of VIP where rules are bent sometimes.
The only US airline allowing pets in the cabin on aircraft with lie-flat seats is AA, and that solely on routes served by their A321T aircraft, and these are all domestic.