Aspirin would not do the job. The seriously ill patients in cytokine storm have so much endothelial inflammation because the virus invades and destroys the lining layer of the blood vessels (where it finds ACE II receptors). This releases von Willebrand factor(VWF) which forms long polymer strands that make for extreme clots. Aspirin is not going to do anything to these VWF strands in your blood stream. It just makes platelets less sticky. Those clots clog up your lung capillaries (looks like ARDS, but isn't), your coronary arteries, your brain arteries, etc. Causing loss of function in all those organs.
NAC (N-acetylcysteine) could help. It contains a sulfur atom that can cleave VWF polymers, and humans tolerate it very well. In that sense, it's kind of an over the counter clot buster! It also helps recharge your glutathione after that protein reduces hydrogen peroxide to water. That makes it a very powerful antioxidant, which reduces the inflammation generated by superoxide and hydrogen peroxide, which are causing the whole mess. The immune cells make these poisons to kill infected cells, but in this case they can make too much and kill us.
Most treatment now anticoagulates with full dose heparin, but some people like EVMS suggest NAC as well.