Any school writes its own complaints procedure regardless of whether or not it is an academy. That is entirely their own responsibility. There is no automatic review.
If a parent appeals to the EFA they will decide whether or not the complaints procedure is adequate.
Ultimately the courts define what a rational person is. Again, no differrent for LA-controlled schools.
Most LAs don't have an ombudsman. Did you mean the Local Government Ombudsman? You cannot complain to the LGO about your school. You can only refer certain specific matters to them - school admissions, school transport, some aspects of SEN, exclusions and tuition at home. Their role in these areas is identical to that of the EFA, i.e. they will only intervene if the correct process has not been followed or the decision is irrational.
If you have a complaint about your children's school that they do not resolve to your satisfaction you may be able to appeal to the LA. However, I repeat, they will only intervene if the correct process has not been followed or the decision is irrational. They will not substitute their judgement for that of the school.
If you want to make that argument about the way the EFA is funded you could argue that the entire state school system is unaccountable. After all, you complain about a school to department of the LA to review a decision made by a school that the LA directly funds. Except that they don't review the decision. They only check that the correct process has been followed and that the decision is rational. And this is, of course, quite separate from accountability for the money involved.
Apart from certain classes of decision where there is an independent appeal (primarily admissions and exclusions), the only people who can review and overturn any decision made by a school are the governors of that school, the Secretary of State and the courts. This applies regardless of whether it is an academy or an LA-controlled school.