My stay at BLT is the easiest I felt for navigating the parks. Bus transportation is easy, the walk is perfect, and monorail usually is not full and easy to board. When we had more resort time, Poly was enjoyable for the pools and my wife seemed to stay in a good mood there ha. I had not stayed at Grand but spent a lot of time there and think with the walkway I would now consider it. Personally walking into MK is my number one thing for keeping my day stress free and would rank that as my most important. I would say BLT for park heavy, GF for nicer amenities but still MK heavy, and Poly as the more resort heavy or other theme park interest. None will be bad but will agree that strollers at Poly are very annoying at times and it is the loudest/busiest of all of them. When I stay at BLT it has things that are not as fun as the others but it never feels crowded in the grounds and you can easily just visit the others for those amenities with little trouble.
PS. I have had my toddler at Disney 3 times before 2 and she will be there at 28 months next time. Each time was dramatically different and the one thing that sucks more than anything is boarding a bus. Monorail was not too bad but it can be miserable boarding last on the loop. I am bias but personally I only desire to stay on the loop or Boardwalk area for that reason. Walking and proximity is so much nicer. I will be staying at AK for first time next time to see the savannah so that will be my first swing at being dependent on buses for 3 days....