I tried to sent this several hours ago but CC wasn’t working for me:
I agree with Publisher. It seems the only downside is the GRE and we don’t know if they were really as bad as you say. The W’s can be explained briefly, just as you wrote here, if need be.
My kid applied to only two grad schools. They had a pretty good idea of funding beforehand, and visited both before applying, met with a potential mentor etc. The important thing is to find the best fit, and that means also finding a teacher/s who fit as well.
By all means apply to Harvard if it is a good fit. You can apply to lots and lots of schools if you wish, but that scattershot method may not be the best approach long term unless you make sure of funding and fit.
Curious: are you a recent grad and do you have any real life/job experience in the field?