I think if you go during one of the "good" times of the year that has been cited, for low crowds AND you get there at park opening, you actually still can achieve that elusive "feeling" of an almost empty park, at least for a little while. Like, last year, I went in the last week of January - if anyone remembers, that week was preceeded by a HUGE blizzard that crippled the entire east coast. I left NY the day before the blizzard hit. Anyway, the first several days in Disney were FREEZING - like it really was 32 degrees Fahrenheit at 11:00 in the morning. But anyway, when I went to Magic Kingdom at park opening , I made a beeline to Adventureland, which is what I always do - my own tradition and anyway, the park in that area WAS beautifully empty-ish, or as empty as Disney World is ever likely to get in it's busiest park. When you combined the frigid temps with the great time of year for low crowds and add in being there at Park Opening and also that I didn't visit where the coasters are first, and with that you get that wonderful feeling of almost having a part of the park to yourself. It's an elusive feeling but I enjoyed it. I passed Rafiki who was taking pictures and though I usually don't bother with that, how could I resist posing with a character when there was no line? So, it was pretty amazing and sparsely attended, at least for a bit in that section of the park.
However, my first week or so in Disney was uncomfortably cold, so that is a major detraction to that time of the year. My 2nd week there, the temps finally normalized. Generally, speaking, I suppose those two weeks, last week in Jan. and first week in Feb. were probably the least attended I've ever seen the parks. Though obviously, not the MOST desirable time of year to go.
Other times I've gone that had a little more crowds than that but still pleasant-ish were the first 2 weeks in Dec. and the first 2 weeks and last 2 weeks in April, when Easter week/Holy Week weren't part of the calendar. I also have gone the first 2 weeks in June and it really wasn't that bad - not all the kids are out of school yet, some are, but it fit my schedule, and while it was less desirable than the other times, due to heat and more crowds, it was still doable.