A previous Conservative Government had a solution for the "what if you don"t like your local school?" problem. It was the Free School policy, but only in its original form, which enabled local parents to set up new schools themselves (whether new places were 'needed' or not). It worked ok in an area near us, where there was a disliked academy trust with a monopoly of 2 secondary schools. Parents set up a new school that was relatively mainstream and normal, so it was more popular and successful than the 2 monopoly academies. The 2 unpopular academies withered on the vine to the extent that they were judged inadequate, and their academy trust was booted out, to be replaced by a more mainstream trust. Since then, they've both improved. Now, that area has 3 good mainstream schools, rather than 2, and a bit more choice.
Of course it's a horribly inefficient process to set schools up in competition like that. Subsequent Conservative governments have, rightly, changed the rules, so that Free Schools can only be set up where additional places are needed. So now, the only solutions for avoiding unwanted schools are the same as they ever were in the past - move house or go private.
But I still don't think this is an Academy v LA issue, because LA schools are just as likely to be unpopular as academies - and, where they still exist, they also have a monopoly.