You need to apply to places that have the specialization(s)within poli sci that you are looking for. If Harvard is clearly a good fit for your interests and you can write a compelling application letter, then sure you should apply.
I recommend applying to more than 6. Not only do you want an acceptance, but you want a good funding package. Your letter writers won’t mind if you apply to more than 6 - I recently sent out about 20 for one student and that did seem like a lot. Although you did say that you wanted to not pay for too many.
Good luck!
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