I did my first trip (solo) this month. And I, too, flew in from the west coast so it was an all day experience for me. I used DME and it was great. "1000's of screaming kids" is so subjective. Yes, there were kids on the bus and some of them were pretty squirmy and one or two were tired so there was some crying.
It didn't bother me, but I know some single adults really hate the sounds of tired or excited kids. If that's a bit of a priority for you, maybe a rental car would be better.
Using DME was easy and fast. Since it was early September around 5:30 PM, there were very few people there - I walked right on the bus feeling a little silly making my way through the winding queue all alone, found a nice seat and waited about 20 minutes while others boarded. Was in the lobby at about 6:30. I checked in quickly, found my room, put away a few things and then took a bus to Downtown Disney. I had a drink (a great, surprisingly strong Margarita) and then a fish and chips meal. I left around 10:30PM and when I got back to my room my luggage was there.
Very nice, very convient. I didn't miss not having a rental car at all.