I've gone to MK 4 years in a row on NYE and LOVED it, and NEVER got "trampled". I do rope drop to close (which the last 4 years has been 2 AM), and when the park opens it's pretty light since the rope drop is so early. I can get on a bunch of rides/attractions in the early hours as walk-ons, and start collecting FPs early. Since I know how to use FP to it's maximum advantage, I actually set my personal record for most MK attractions ridden in an operating day on a NYE. The LONG operating hours make it possible to do more, and FP cancels out much of the time that would be lost to crowds if there were no FP.
On a typical NYE, I can do every attraction in the park at least once and have time for night encores on SEVERAL favorites to close my night and ring in the New Year with a bang! And I still have time to experience the entire Christmas entertainment package - I see the Once Upon A Christmastime parade TWICE, the Celebrate the Season Castle Show, Holiday Wishes, and the Fantasy In the Sky NYE fireworks display to ring in the New Year.
After the fireworks, much of the masses and stroller set makes a beeline for the exit, and by 1 or 1:30 AM, attractions that had insane standby lines in the peak of the afternoon are often walk-ons. Of course, I always have a FP handy, just in case they are not!
I won't lie, though. There are SOME things about the NYE experience at MK that take getting used to. The pathways can get EXTREMELY congested, particularly in the Hub at nightime as many people get in one spot and stay there all night for the parade, Castle shows, and both firework displays. But since I usually watch the parade in Town Square and/or Frontierland, and prefer to view the Fireworks in the center of Fantasyland, this affects me to the minimum.
Also, the constant litany of people blowing horns all night from as soon as they are passed out in the afternoon can take some getting used to, as does the contemporary beat-heavy "Todayland" music and loud DJ chatter that replaces the traditional music loops on the parkwide audio system, and this kills the illusion of being in the western frontier, an exotic jungle, or the lands of storybook fantasy and futurism. Due to this radically different audio, I usually try to do at least one other full day at the MK each year when it has the long Holiday hours and Holiday entertainment package, to get a more traditional MK experience combined with the long hours and Christmas overlay.
Those are my only NYE "pet peeves", though, and other's mileage may vary. Now that I know what to expect, these things are less of a shock, and overall, I find the experience very satisfying, particularly being able to ring in the New Year in my favorite park with Mickey Mouse and the gang counting backwards from ten at midnight, the amazing Grand Finale of the Fireworks, and then ringing in the New Year on all my favorite attractions!
After the park is officially closed, I enjoy lingering on Main Street for yet another hour (as if 19 hours in the park wasn't enough!), enjoying the Castle Lights, Christmas Tree, Main Street Lights and other Christmas decorations, and of course the beautifully sublime Main Street Christmas music loop (which has thankfully reclaimed it's rightful spot from the loud DJs by this time!) And the atmosphere on Main Street at this time is very peaceful, serene, and relatively uncrowded, with scarcely a hint of the chaos it must have been like a few hours before!
And then after watching the MK lights fade away from the upper deck of the ferry, I like to sleep in, VERY late the next day!
I am already looking forward to this year's Holiday Season and NYE! Only a little more than 3 months to go!
EDIT: Oh, sorry, I see now you specified Epcot. Well, the general idea should still be the same, that if you get there at rope drop or very early you will get a good head start on the crowds, and not have to worry about getting shut out if the park reaches capacity. And you should have no problem doing any attraction you want that day if you use FP.