i>In theory, I have far more control over my government than my insurers.
You can change Insurers, but not really your Government.
So, you're one of those people who believes you can jump straight to a "free market solves all problems" in insurance? A field that both has incredible scale effects leading to a natural oligarchy and, by necessity, is highly regulated? Both of which lead to incredible barriers to entry.
Further, there is little reason for any insurance company to deny themselves this information. First, they seek an advertising, not informational advantage: they all use similar/identical algorithms from the same consultants already. Secondly, in addition to whatever you can gain from serving low-risk individuals who object to monitoring in the pool, would be more than offset by the adverse selection pressure that pushes all high risk candidates into that pool.
Lastly, while I questioned the free market claims, that was a precursor to saying that you are free to emigrate. But unlike a company, where your only choice is to patronize them or not (or there is no choice if you want to drive a car/survive an illness/etc. except for among near identical actors), in the case of a democracy, you can actively work to change things. I'd be hard pressed to find a boycott that worked "well", where I will define "well" as achieving a tangential goal in a timely way through the loss of income to the company. I distinguish this from modifying policies to avoid bad press by a boycott being published, to boost shareholder value by increasing intangible assets, etc. However, I can indeed point to many changes made to a democratic country because some group decided they cared about it.