You’re describing something that’s been a nuisance for everyone who has had to deal with this problem. The plastic isn’t heavier than water, so it doesn’t necessary sink into the bottom of the water heater.
As you’ve seen, it readily flows out of the tank to clog your aerators. Even if you could get someone to vacuum the tank, it wouldn’t necessarily get it all.
There are no guarantees, but if you replaced the dip tube with the flush kit we sell that has a curved dip tube and a ball valve drain, you’d have a better chance to flush the stuff out. The tube would direct the flow better and the ball valve would give you a larger aperture that could also easily be unclogged.
Probably, though, if you get that old tube out of there and flush the tank thoroughly, the problem will subside.