This is one of the things I noticed when running 5e as well. I think the reason is the faster combat that lets you run through that mini-dungeon in an hour or two instead of more than a whole session. This leaves more room for story, which makes it more important to the players and makes the leveling process less important.
This is my experience as well. For instance, the way I am doing initiative now is to have a cardboard triangle for each monster with a number on it, and one for each PC with the name on it. We roll for initiative, and then I sort the cardboard triangles in a line in front of me, first initiative to my right, last on the left. I use a token to show who's initiatve it is. What this accomplishes is to track initiative so everyone knows who's turn it is and it's really easy to remove dead monsters from the list and so on.
Compare the Basic rules to the PHB to get what I'm talking about.
