I always play on the current day. I've only time-traveled once and that was to make a villager move out so I could adopt my all-time favourite villager Walker.
In my main town, yes. It never leaves the date. If it does, it's only to get a villager out asap, but it's rare. If I do, I always make sure to return it to the original date and time. In my cycle town, it's stuck on the 30th of July (2016) due to my cycling routine.
I rarely play on todays date I just like to dig up things and get my town as developed as I can so I can update my dream address and make my town Pretty :0
I used to play day to day but ever since i lost my ds and hot another one, i have been trying to catch up. So I have been tting all the time to move things along. But once i get my town looking okay I'll probably slow down a bit. Especially since school will be starting soon