Every year I play in 3 league for sure: one 10 team, one 12 team, and one 14 team.
In the 10 team league I have never missed the playoffs.
In the 12 team league I am always at least on the bubble but don't always make the playoffs.
In the 14 team league I alternate between having great years and poor years.
So assuming that my skill doesn't change between drafts, one might say luck plays a factor more with larger leagues. Of course, what could be equally said is that I'm not very good at the #100-#140 picks, which are amplified in larger leagues because those are higher draft picks, but minimized in smaller league as they are low end draft picks.
What I've often believed however is that my earlier drafts tend to dictate what players I take in my later drafts. I know I shouldn't do that, but it happens anyway and I do have 3 or 4 players on every team. Am I reaching for those guys? Probably. And again it doesn't affect me as much in the smaller leagues, but in larger leagues reaching becomes a bigger problem. And, of course, my 14 team league is the last draft I have.
So in the end, I think larger leagues need more skill (and more discipline) for drafting, and more luck/great waiver wire pickups during the season. The luck however can be minimized by a great draft.