And all those Hawaii chain islands are in a no-go zone, with no fishing or anchoring permits issued, as of 2000 presidential edict. However, if you were looking for a shallow spot in the Midway Island area of the Pacific to drop anchor, there are several seamounts within 200 miles outside the usa eez out there which are 60 to 500 ft deep. Off the seamounts the depth goes to over a mile. Many of the mounts are un-named, but there is the Milwaukee seamounts out west of Midway, outside usa EEZ. There’s one seamount just outside the usa eez west of Midway, ~300ft deep and one NW of Midway at 300ft. Maybe you cannot build on them, but praps you can drop anchor, the Japan and Taiwan have already dragged all the coral off them.
I don’t believe you have a mental grip on how much garbage there is out there, or what form it is in. There isn’t enough visable plastic floating at the surface to build a net big enough to trap the visable plastics semi-floating in the top 50ft of water. There’s far more plastics you cannot see, estimates based on tests are if you can trap most of it, then you are also removing all the plankton from the ocean. The amount of plastic per volume of water is so low, you will burn more fuel to recover it than if you used the fuel to make plastic. Plus, the big problem of plastic is the UV disintegrates it, whatever you make from the plastic will rapidly deteriorate in sunlight.
The trawler that was periodically used to remove washed up large plastics from the islands was taken out and sunk a few years ago. Another USCG ship was “retired” last year in Hawaii, and i lost track of what happened to it, it will most likely also be sunk in deep water. I think you have a better chance of grabbing the MV Lyubov Orlova, before it is sunk or cut up, and keeping it, than making any platform out of floating plastics in any ocean gyre.