The reason I specified if it could be done on a non profit basis is because the money comes from the public purse.
Historically the government has never chanted that it got "Best value for money" on anything. It just touts a figure (£50 billion HS2) and there you have it. Everyone expects a slice of a cut of £50 billion regardless. So the mentality is wrong from the very foundation of the get go.
If the mentality was, lets secure the land, lets get the legal stuff out of the way and then set off with a mentality of building the thing on a non profit basis (At least the labour, pay a competitive wage but no more) then I think you would have a professionally built railway made cheaper.
If a railway is built on the model of say a pint of beer sold in a pub, then the raw product costs pennies but the end product costs pounds. And I think thats where the problem lies in railway construction. If railways were cheaper, I think it would be a lot easier to lobby for it.
The terminology you are using is complicating things.
The government isn't 'non-profit'.
Public sector is generally funded by taxation in absolute terms, and through borrowing. The 'lower' tiers of the public sector can be profit making but any profits/dividends go to the public purse or are reinvested. There is no 'profit extraction' to private shareholders.
Where service provision is included, it has to provide value for money. That's not the same as making a profit in a narrow financial sense. But the cost of providing that service should bring financial and other e.g. social benefits. There is typically expected to be net positive case (even if it's a financial loss/subsidy to the public purse).
Private sector is funded by equity (shares/equity issuance, private investment) and borrowing/debit. Profits go to shareholders.
Non-profits is usually the term used to describe various forms of charities. They either work via donations. Or where services a provided via fees on a largely cost-recovery basis.
What I think you're really doing a less than great job of explaining is the spectrum between:
nationalisation - paid out of the public purse - there is no private profit extraction...but there is a school of though that implies the government isn't good at this stuff relative to private operators, the market and competition
private operators - would only ever build stuff with a clear profit incentive. So huge parts of scotland and wales etc. would never get rail as it would be loss making
hence the government comes up with private/public partnerships - where the private sector inject money in term for some of the rewards as well. These can go badly when the private sector takes the profit but the public sector takes on all the risk.
As Bald Rick alluded to - people don't do anything for free. Major infrastructures projects are expensive and it's a long time. Income can lag expenditure by a decade or so given the gap before fares can start to be charged.
Private companies are under a legal duty to act in the interests of their shareholders. They won't work for cost recovery. They would never get equity investment until shareholders knew they would get dividends or be able to sell shares for a hopefully higher price.
And a 'non-profit' build isn't necessarily cheaper. You seem to be thinking merely in terms of e.g. materials and labour in finite terms. In practice, a huge part of the costs can relate to bad project management and overrun - things the public sector is more prone to.
In a Utopia say a project cost 5bn.
A public sector build might cost 7bn - bad project management, a few strikes, a couple of years delay when there was a change of government and costs went up etc.
A private sector build might cost 4bn. But a 5 years later it's discovered that there was some cutting of corners to deliver on time. So another 2bn is spent making adjustments. Then it seems better value for money for a while. But then maintenance costs go up, because the contract is about to end and the private sector refuses to buy new whatevers as it's not thinking in the long term.
