That old question comes up from time to time, so I thought I'd add my thoughts.
I took my Kona Heihei to the Mash-up at Glyncorrwg last year. I thought "hey, it's only a trail centre and I'm doing the Enduro so it'll be nice and quick on the Kona". Well, I was wrong. I got pummelled on the downhill sections, and my upper body was really tired by the end. My times on the DH AND singletrack sections were crap. I was really wishing for my 5 by the end of it. And I would in no way describe those trails as rocky. In fact, I think they're particularly smooth.
I've been taking my 7" travel Patriot to Cwmcarn lately, and it's absolutely fantastic. I mean really really brilliant. I just get faster and faster on the descents.. an old trail that I've ridden dozens of times a year since it opened is giving me more challenges and almost delirious amounts of grin factor - and it's because the bike is just begging me to push harder and just go faster and faster all the time. Is it the amount of travel or the geometry? Both, I am sure, but travel's a factor, and you don't get 66 deg head angles on bikes with 80mm travel.
So.. of course you don't need travel at trail centres, or anywhere, but I know which bike I'm taking next time