Well, maybe. Legislators are really variable on constitutent service, but many will (or rather, will allow their office staff to; direct, even by something going out under their name rather than a staff contact, is less common) at least gently probe on behalf of constituents on state service issues, and if a high enough volume is involved are more likely to get personally involved.
But any response that does happen is likely to involve butt covering and blame redirection from virtually every step of the management chain, just like when an issue hits the news media, so on issues where you haven't exhausted other avenues short of lawsuits, it's probably not the step to reach for the way escalating to a manager would be. I’ve definitely been involved in issues where premature escalation to a legislator delayed and/or made the resolution less favorable than it probably otherwise would have been because of the degree of attention from management, legal, public and legislative affairs, etc., that the escalation triggered.
It makes sense in the particular circumstances described, though, it's just the analogy that was flawed in a way which might be misleading as to how that generalizes to other situations.