I'm not from LI (really no where close), but facing a two day drive, I'd probably leave around 8, no later than 9 in the morning.
I don't like scheduling my "midpoint" stop. What if traffic is clear and I'm doing better than I thought? I could get another couple of hours in. By the same token, what if there's an accident, construction, or something else that slows me down? Then I'm trying to "push through".
If it looks like I'd need a stop, I'd have someone else in the car get out their phone and start looking at maps/reviews and figure out where to go a couple of hours before I think I'm done.
ETA: It's still not hard to use my plan if I need to allow for traffic in a certain area. If I get further than what I think on Day 1, I can start later on Day 2. If I don't get as far, I can start earlier on Day 2 to still hit the timing I want/need.