Writing to follow up. In case someone hasn't read my other posts in this thread, I am on my 3rd 6S Plus, 64GB, Space Gray, no idea what chip or build week. I ended up fixing the lag on the first two, but they were both returned for other reasons.
iPhone #3 was the most stubborn of them all. Here's how it went down.
9.0.2 out of the box.
Set it up temporarily in the store just to get calling and texting going, and drove on home. Went straight to an iTunes update, restore and set up as new. No dice. Lag and homescreen freezing. For giggles, I decided to try the lag-free working iCloud backup I had from phone #2 that I made earlier in the day. Upon rebooting from that iCloud backup, the phone got stuck on a full blast brightness white screen with black Apple Logo in the middle. Stayed that way for 15 minutes and the phone was pretty hot (probably from the screen brightness being up to 100% for so long). I two-finger reset it, and when it rebooted, it went into the setup screen as it should have done automatically. Went through the setup to get it up and running, and when it came time for the apps to start downloading, I put the phone down and left it alone for an hour or so. I came back, apps were all there, lag and freezing were totally there. At this point I said, "one more shot at this and I'm done."
It was pretty late, and I stayed up a little late to do it, but that was part of the plan. I again did an iTunes restore and set up as new. After it was done, I purposely didn't check it for lag or freeze. I use iCloud Photo Library, and I know that can take a while to re-sync. I turned that on, plugged my phone in to the charger, and left it beside untouched overnight. When I woke up in the morning. No noticeable lag, and definitely no homescreen freezes. I've since installed all my apps and set everything up the way I want it, and still no lag or freezes. I did this last restore on Friday night, and used it all day Saturday and Sunday with no issues.
I'm not into jinxing myself and saying that it's all in the past, but this really has seemed to have done the trick. I also remember with phone #1, I did the same thing where I restored from iTunes, went through the initial setup screens, and then let it sit overnight, and I also woke up to a lag free phone on that occasion.
So, long story short. Give it a shot if you haven't already. Restore it through iTunes, run through the initial setup, then put it down over night. If you use iCloud Photo Library, turn that on and let that do it's thing too, because that can take a while and potentially cause extra lag.
9.1 just seems really temperamental. I ultimately got rid of the lag on all three phones, but the path to get there was a little different on each.