I am a couple of sessions in now. I decided to run Cold Hearted Killer.
As predicted by many, my group didn't actually go deal with Cold Hearted Killer straight away. I started them in Easthaven and did the Séance in order to give them some hooks to work with, they then learnt about the killer, and then I gave them the Toil quest for Easthaven.
They decided that Toil and Trouble was the most urgent. They played this quest smart and didn't have too many issues for a level 1 group. Although a newbie group could easily TPK here, they didn't. The dire wolf, hag, and will-o-wisp was pretty easily dealt with. They avoided the frost giant skeleton. They did set off the water weird and struggled with it a bit at level 1, but overcame it in the end.
After completing the quest I gave them some opportunities to find out what the assistant at the Town Hall was up to (Prudence), and they managed to figure her out and prevent the Zhent's from stealing the cauldron. Capturing Prudence also gave me an opportunity to tell them about Revels End. However PC's being PC's they completely ignored the Lady on the Lakes warning about the Duergar.
All in all I think with a little bit of DM guidance and having a group that has some sense about them, it's very easy to avoid many of the pitfalls that people have been worried about with starting this adventure. So far it has been really fun.