I totally agree with your thinking Holly - if you have time to spend in your hotel in New York for more than just sleeping, you are missing out on something !
However, I suspect you will find it difficult to get a decent, centrally located hotel for €100 per night. I stayed in the Crowne Plaza in Times Square on my last visit. It was very central, reasonably priced (by NY standards) and clean.
Have you thought about looking for something smaller like a guesthouse in one of the neighbourhoods like Soho or Chelsea ? You might get a better taste of the real New York than in a big hotel chain.