The bandwidth issue is a small obstacle for a potential competitor (solvable by venture capital) relative to the issue of video copyrights and offensive content moderation.
The VC issue is interesting. It would probably cost... hundreds of millions of dollars in capital over a few years to compete with YouTube properly (requiring HD, long length video), given the time it will most likely take to get up to speed on revenues / build a functioning business model, and the speed at which adoption often happens these days with successful products (wake up tomorrow with 50 million users wanting to stream your brains out).
We live in the age of pervasive offensive Internet content. Reddit is half offensive content, and then there's Tumblr and Snapchat, which thrived partially due to 'offensive' content. I view that as a trivial problem.
Whatever replaces YouTube, will likely be built on the back of sensationalistic offensive content. Nothing sells like sex / sex appeal, and that will never change. Indeed, people get accustomed to it, and then it takes even more dramatic content to get that sensationalism buzz.
Assuming you mean sexting, I think the percentage of sexually explicit images sent on Snapchat has been wildly exaggerated — by tech journalists looking for clicks, and by older folks not in Snapchat's target market who don't really get the point. Snapchat's appeal for everyday, non-sexual use seems to go over the heads of most tech people I know, but has been described eloquently by, e.g., Nathan Jurgenson[1] and Dustin Curtis[2].
Porn on Tumblr and sexts on Snapchat make for interesting headlines, but I'd hesitate to credit either as a major reason for the platform's success without actual numbers on its prevalence. To my knowledge those numbers don't exist.
1. http://thenewinquiry.com/essays/pics-and-it-didnt-happen/
Their conclusion: a smaller percentage of Snapchat users use the service to sext than text-messaging users.