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Mostly expense and difficulty.
You can't just throw 50 types of plastic into a single vat and expect to get usable end product. Not all plastic types are compatible, and there are also safety issues involved with the chemical reactions, so on, so forth.
Plus actually gathering and filtering the plastics is a gargantuan undertaking, especially if you want to put forth any effort into not wiping out what marine life manages to survive there.
Most of the plastic is in such small pieces that you'd need something finer than cheesecloth to do a descent strain, which means you are wiping out algae and other microorganisms that are the most basic parts of the ocean food chain (and the source of most of the planet's oxygen supply as well).
In short, sadly, it's past the point where we could just reclaim and recycle. If only the folks 20 years ago had listened. Now, all we can do is try to stop it from getting worse. Right now the problem is growing far faster than any means we have of cleaning it. It's like trying to shovel a sidewalk during white-out conditions.
The 'reclaim and recycle' idea will be the eventual solution, but it will take time, and before it can work, the dumping has to stop. Which does mean that there must be some restrictions placed on businesses that otherwise will try to shore up their bottom lines by taking the cheap methods of disposal.
And it also means that littering laws will have to be enforced.
One good thing you can do for the environment is purchase a taser, go to the beach, and target everyone who leaves their trash (including cigarette butts) anywhere other than an appropriate receptacle.