A state-run enterprise is to me not preferable over a healthy market, but it is almost always preferable over a private monopoly. As a citizen you have no leverage over a monopoly, whereas a state-run enterprise still has a democratic feedback loop.
Also, the Republican small government platform goes much further than just railing against government-run enterprise: it essentially states that all government intervention in the economy is bad (both morally and from an effectiveness standpoint; the effectiveness angle is crucial to make their platform appeal to enough people).
In the case of ISP monopolies in rural areas leading to low quality and high costs, I think it doesn't seem likely that the free market will solve this problem by itself any time soon. So politicians without a small government ideology might consider methods such as good old breaking up monopolies or other antitrust measures. Of course, this can not be done at a local level, and if small government ideologues are in power at the state level, this approach is dead in the water.
So if you're a local government and you want to attack this problem, what are your options? Municipal broadband is the only approach I can think of that's within the reach of a city council. So yes, it's a band-aid, but it's the best you can do as long as the state-level is run by small government ideologues.