The fragmentation is intentional, on the part of the content owners. Believe me, everyone knows that a lot of people want a single streaming service with all content. It's just not what copyright owners and ISPs want.
Let's say Netflix suddenly had the rights to stream all movies, TV shows, and live events, and became the service that pretty much everyone used. Even if they raised prices quite a lot, people would still sign up for it. However, a company like Comcast would then be relegated to being a "dumb pipe". People would still pay them for Internet access, but they'd lose most of their revenue for cable TV or streaming services of their own. It's actually in Comcast's interest to keep streaming services insufficient to replace cable, so that people will continue paying for cable.
A company like HBO wouldn't like it, either. They could continue to make money by licensing their original shows, but that's not the only way that they make money. Even for their streaming services, they make money by bundling a bunch of content and charging more per month than you would probably pay for their original shows. However, a decent chunk of income comes from deals with cable providers, which would dry up quickly once everyone had moved to Netflix.
Even the networks and production companies that produce shows probably wouldn't like it, for the most part. Right now, they can license the same show or movie to Amazon, HBO, iTunes, Hulu, Amazon Prime, and any other services they like. You're forced to pay for a bunch of different services, and they get a small cut of each of those services' incomes. If the number of services were cut, they might also make less money from that one service than the aggregate of all of the other many services you're paying for. After all, you may be subscribing to some of those services for only one or two shows (because a fair amount of the content on each is available on the others), but the extra money still gets split among content owners.
And all of that still hasn't touched on the fear of one company controlling the whole market of video distributions. If Netflix were to get access to all content before the other streaming services, then they could become a de facto monopoly, and control distribution for all the different content owners. Even if all of the streaming services suddenly had access to all content, they would lose most of their marketing leverage. They would only be able to compete on things like the quality of their apps, the quality and bandwidth usage of their streams, or price. You and I might think that sounds great, but it's not really what the industry wants.