This gives me an idea. Introducing, the Brennan airplane (see: 7:20 https://www.youtube.com/watch?v=kUYzuAJeg3M)
Banking the plane is a nice gesture to help everyone experience a special occasion.
So I was on a flight from London to Vancouver, which had us approaching Vancouver from the north. It was December but a sunny day. Half the passengers were on ski holidays. We had to delay our arrival a while so the pilot said he was going to bank the plane to the left and do a circle, which would put the Whistler-Blackcomb ski resort in view out the left side. In unison, seatbelts be damned, everyone on the right of the plane got up and moved left to get pictures. About half-way through the turn, the pilot came back: "Um, we felt that up here. When you return to your seats can you please do so slowly, at least slower than how fast you moved to the left." Pre-9/11 was a simpler time.
I doubt everyone moving side to side when the plane isn’t taking off or landing is going to automatically cause a crash.
Side to side weight balance of passengers is not a major factor in any plane I have ever read the manual for. They are too close to the centerline to impart much force. The pilots might notice it and trim it out, but I doubt they would have to do anything beyond that. Fore and aft balance is critical, but still unlikely to be an issue with passengers at cruise speed since the elevator has a lot of control authority at 500mph.
A 737 can fly fine with a completely empty wing tank on one side and a completely full wing tank on the other. The wing tanks of some models can carry 4,500 kgs of fuel in each side, so that amounts to a 9,000 kg imbalance outboard of the fuselage. That is the equivalent of about 111 "standard" passengers' weight on the wing that can be handled with trim alone, and 737s are capable of safely landing in that configuration.
Boats are different because as they heal farther and farther the righting moment diminishes (for conventional monohull powercraft at least), and the passengers are able to get all the way to the extreme.
Planes on the other hand have the same "righting moment" no matter how far over they bank since it is a function of airflow over control surfaces rather than buoyancy and gravity.
Sounds like it might be something that they should tell people about, I guess they don't because when there's enough people in an airplane people either tend to (1) behave according to rules or (2) just not be a significant amount of people to cause the plane to {{do funny things}}.
...but in any case, I can imagine being a pilot and hoping/trying to/prayin for people to stay in their seat regardless of whatever is happening outside...
...must be a scary experience...
If people think they won't get a chance to see, they might all crowd to one side of the plane, causing enough weight imbalance so as to cause a crash.
It's a big problem on boats, and there are countless stories of someone on a crowded boat seeing a dolphin, shouting about it, everyone crowds to one side to get closer, and the whole boat capsizes.
Regulations now require that boats stay afloat if everyone stands on one side, but the regulations aren't perfectly adhered to and that still doesn't prevent people crowding to one side enough to push/knock others overboard.