OK, I think it's hard to advise, as it really depends on what your actual school choices are like. But a couple of thoughts that might help.
First, there's no way I would be paying for a school that infuriates me that often! What's wrong with it? You call yourself a nuisance parent - does that mean you have ridiculous standards and would pick holes in any school, or do you think the current school is actually 'failing' regularly? If you're someone who's always going to find fault anyway, then maybe you're better doing that in a school where you're not paying for the privilege. OTOH if your infuriation is the result of poor 'customer service' from the school, then you may well find this is worse at a state school, where budgets are tighter and staffing is under more pressure.
Second, my experience of the parental involvement required at private secondary is pretty much the opposite of your experience (I've only ever done private secondary, not prep). I never have to chauffeur my children anywhere - that's one of the joys of their school. School transport is provided to all sports fixtures, trips etc. School days are longer than state, which frees me up to do my job better. There's endless wraparound provision if needed (eg breakfast, dinner, music practice rooms, library, common rooms etc which are open early until late). Not like some of our state schools which kick out at 2.30pm and have one bus home (we're quite rural too).
Academically, it will depend on the schools available. Ours is very strong, though we also had very good state options. So the private option edged it on academics, but only just.
As for cheaper vs more expensive schools, the question is not only what your options are, but what you're looking for as the advantage of private. Arguably, if your state options are good, then the advantages of private are the fantastic facilities, the extra curricular clubs, etc etc. Those things come at a price, so the cheapest private schools are less likely to offer lots that a good state option doesn't. But if your state option is awful, and all you're really wanting to pay for is the option to escape it, then a cheaper school with fewer bells and whistles might appeal.
What it boils down to is, you need to decide what's important to you in a school, and look at what your state options are like, before you can really decide. But if you're as high maintenance as you imply, then you might find a state school frustrating in terms of communications, organisation, facilities and so on.