Rather than go over the specifics yet again I'll look at the general argument. Your viewpoint seems to be that government should not intervene and should let markets look after themselves, with regulators regulating where necessary.I'm doubtful whether increased public spending of any sort helps the economy, as it increases the national debt, which increases the ongoing burden of interest payments.
Fair enough, though I don't share your enthusiasm for Ofsted. Of course there are dangers as you've pointed out, but the alternative to action is inaction and if you leave a depressed economy to cure itself that can take a long time. Yes the government sometimes gets ripped off by private sector contractors and it's important to watch that closely. We're proposing a policy that stimulates employment, keeps money circulating in the real economy, and achieves environmental objectives.
But let's assume you are going to have a spending spree of tax payers' money (or stimulus as some call it). The least bad way to do it is to reduce taxation, reducing national insurance perhaps. This sets more money free to earn greater returns.
The worst way is to spend the money on some government scheme or other. Governments shouldn't be in the business of picking winners or funding enterprises. If that does happen, then the business in question will become addicted to state aid, and forget about providing goods and services to customers, and a downward spiral ensues.