One possible downside - I wonder if we'll suddenly find we can't get hormone therapy treatments due to over demand? There is already an increase in demand due to it being temporarily used to delay treatments.
I think they would need to use the fast acting ones with patients already ill with COVID-19, which are Bicalutamide and Degerelix. I don't know if Bicalutamide would work, being an anti-androgen - I haven't looked to see if the pathway to TMPRSS2 suppression is known yet. The LHRH Agonists (Zoladex, Prostap, Decapeptyl, and in the US, Lupron/Eligard) take best part of a month to start working, which would be no good in that scenario, but fine if you're already on them.
Of course, it may be that even fast testosterone suppression doesn't suppress TMPRSS2 quickly. These are all unknowns at the moment.