If it costs money to do something and you hand it over to the private sector it will cost money plus profit to make it therefore more.
Wrong. If that were true, the USSR would have economically destroyed the US. That's just one of millions of examples. It's not the case that *everything* is best done by private enterprise, but if the primary goal is to serve the customer at minimum cost, competitive private industry is the absolute best way we know to achieve it. Yes, companies need to generate a profit, but that profit is almost always dwarfed by the opportunities for reducing costs by being more efficient.
In a competitive market, finding a way to reduce development and production costs increases profit in the short term, which is why companies work really hard to do it. Then in the longer term competitors adopt the same cost-reduction strategies (or better ones) and lower their prices in order to take business from their competitors, lowering the cost to buyers. At the same time, competitors look for ways to make their products more appealing to attract buyers. This virtuous competitive cycle in nearly all cases results in lower prices for better products because -- and this is the key point -- the need for improvement is relentless, never-ending.
Government agencies have different incentives. Not that government employees can't be interested in efficiency, but the organizational incentives are not focused on minimizing cost and maximizing service in order to maximize competitiveness. There is no competition. Government organizations are focused on compliance with the regulations that define the reason for their existence. If the required duties are performed within the funding allocated, they've met their goal and there's no reason to try to seek better ways to do their job.
Note that in both cases I'm speaking of idealized models. Many markets are not competitive (for example, I'm not sure a truly competitive market in health care can exist, because the complexity of the products and services exceeds the ability of consumers to buy intelligently, plus there are serious moral issues around tying availability of care to ability to pay) and private employees have an individual motivation to sit on their hands as much as possible. Many government employees are focused and driven and just as relentless about improving what they can as any business. But on the whole, results align with incentives and private enterprise has an incentive to improve that does not exist in government agencies, even those with open-ended missions.
There's a place for both private and public sector organizations in fulfilling social goals, but correctly allocating responsibilities to them is complicated and requires a deep understanding of what each does best and what each does poorly. Incredibly simplistic views like yours are not and effective guide.