Not OP, but I just recently watched Dances with Wolves a week or so ago and had almost an identical thought to what OP posted. I have some modern favorites of course, but it seems like I enjoy my modern favorites because it is a "mental tease". They don't really illicit emotional connections though. However, it is totally possible I'm just watching the wrong movies. Some modern favorites are the new Blade Runner, Ex Machina.... there is a big list.

There are of course exceptions. I also recently watched 1917 and I thought that was one of the best films I'd ever seen. In addition to being an action film it definitely felt emotional.