I don’t think 1e and 2e had starter sets right out of the gate (probably because basic was still out and they assumed people would start with basic). After initial release, 2e did a bunch of stuff that could be considered starter stuff. Many already mentioned, but also the class specific adventures and a few others.
Can’t speak for 3e or 4e, but the best one in my opinion is the Mentzer set. That literally walked you through and adventure as a new DM. LMoP doesn’t even really do that.
I also liked In Search of the Unknown because it was like a training wheels adventure, giving you the outline but allowing you to flesh it out yourself.
I know a lot of people give KotBL flak for being a dungeon hack, but I think that does it a disservice and is a missed opportunity. The surrounding areas had details that allowed sandbox exploration, and the keep offered many intriguing possibilities. The caves themselves introduced factions and alliances between monsters that hadn’t been done before. Creative players would leverage this to their advantage.
Needless to say, I’ve played and run that module many times, and some of them ended up with the PCs allying with certain monsters to take over the caves, then attack the keep itself ala a Trojan horse.