Yes, they are standing to board. Because the NYC train station is under ground (tunnels are needed to get under the river, the platform is also under ground like a subway. The platform for the tracks are very narrow. So a train will come in, it will disembark all passengers and change crew. During this time above ground passengers get in line to present tickets. They then walk downstairs to the platform and board the train. But because the platform is underground and relatively small, it is not a comfortable or safe place to wait to board while guests disembark.

Platform for amtrak at Penn Station/Moynihan:

I think that is correct about consider Amtrak like an international train like SNCF in France to TGV. It's passenger focused, but generally for longer trips with people carrying luggage. We also have regional/commuter trains for rides less than ~90 minutes meant to get people to work in the city.

For seating, it's somewhat limited because of the size of the station, but it's also right in the middle of new york city. If you are looking to sit for an extended period of time and the weather is nice, there's public seating at bryant park or the squares or blocked off seats on broadway outside of times square. But NYC doesn't have a ton of space, and the train station is designed to prioritize the train passengers.