You can directly search NT database: which includes everything and play with -v and -b options (to a very big numbers) to export as many similar hits you would expect from different genomes.

I don't think it sounds reasonable at all. Remember that BLAST is local alignment, you'd find a plethora of sequences with some small region of similarity, and in the next iteration, you'd find sequences that match completely different parts of the first round results. If you are looking for remote relationships, you are better off using a more sensitive alignment method, or stochastic models like HMMs.