In my case, Senator inquiries have had a direct and immediate impact on USCIS to pick up and act on my case both post-NOA1 and pre-RFE, and post-RFE. Otherwise, I can almost guarantee that my case might still have been languishing in some cold, dark corner waiting to be opened/touched and even after RFER, it might not have been looked at until close to the cursory 60-day post-RFE published timeframe.
Specifically, Senator Ed Markey whose office has been very welcoming, friendly and very responsive (He has my vote, my money and my network forever).
My rep Katherine Clark's immi assistant simply refused to even send an inquiry saying she has tons of immigration experience and that it never works out until the USCIS published times are way past.
My other Senator Elizabeth Warren initially refused with a similar canned response, and just now sent a response asking me for a physically signed auth form so they can send in an inquiry (6 weeks after my request to them), but thankfully I don't need them anymore.