There's the large assumption that the flag carrier country will pay.
Realistically, once this becomes a strategem, and a known one, anyone with a boat can enter your waters, hoist the jolly roger, then say "Yeah, take me into asylum." And poof, your immigration procedures evaporate.
You should do some reading up on what is involved in applying for and receiving asylum in the US. Your post indicates an inaccurate understanding of what is involved and required for an asylum seeker to receive legal asylum.
I assume it is like immigration. In one sense, it is a long legal process with a variety of steps to ensure this person is serious. But in a more practical sense, it just involves crossing a border and having the phase "illegal immigrant" removed from the vocabulary.
It would also have the nice side effect of dissuading countries to lend their flag to shady companies all over the world.