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Mistakes can sometimes be excused if lessons are learned. And there's plenty to learn from the rebuilding of Wembley Stadium, now under construction, at very long last. I'm not talking about lessons to be learned by the Football Association, whose project this is. If the FA had decided to demolish the 80-year-old and distinctly seedy stadium and had decided that the project should be funded entirely with money provided by the FA – either its own cash or borrowed money – then that would have been entirely its own business. That the project is years late, that it is far more expensive than originally forecast: those should have been matters for the FA only.

The problem was that others got involved – indeed, volunteered to get involved or downright insisted on getting involved. Sport England provided £120m of lottery money, on the clear understanding that the new stadium would be dual-purpose – for football and for athletics. The House of Commons culture, media and sport select committee, in a report this summer, castigated Sport England for its role; Sport England has stated, contritely, that it is examining that role. Let's hope the mishaps connected with its part-funding of the stadium will never be repeated.

But I am not even talking about lessons to be learned by Sport England. No. What I am talking about very specifically is lessons to be learned by government – and the omission of the definite article before the word "government" is deliberate. Government involvement in this non-government project began with the Conservatives and was continued when Labour took over in 1997.

Wembley Stadium was none of any government's business. The stadium was built in 1922-3 with no government involvement whatever. Unfortunately, former culture secretary Chris Smith seems to have been misled into believing that because Wembley was popularly known as the national stadium, it really was the national stadium. It never was. But because of this misconception, Smith started laying down rules about whether the new stadium should be dual-purpose or not. Instead of the FA telling him that it was none of his business and that he should run away and play (only not in the stadium) it heeded his insistence that it should be a football stadium only, despite the lottery grant being specifically for a dual-purpose ground. It therefore accepted delays, redesigns and the decision – based on nothing but wishful thinking – that athletics should be accommodated not at Wembley but at a purpose-built facility at Picketts Lock, in north London.

Never, never, never again should government get involved in such a project in any way whatever

When Tessa Jowell succeeded Chris Smith last year, she immediately cancelled Picketts Lock, acknowledging that it had never been a runner (sorry for these unavoidable puns). But then she, too, started getting involved with Wembley – once more dual-purpose – and this week she was due to make a statement about it in the House of Commons.

I hope that this will be the last Commons statement any government minister ever makes about this project. From now on, the government's only concern with Wembley stadium should be the properly controlled and supervised expenditure of the £20m subsidy it is providing – unwisely, in my view – for infrastructure. There should also be stringent supervision by Sport England of the use of its £120m lottery grant. Not one single further penny of Treasury money should go into the stadium. And never, never, never again should government get involved in such a project in any way whatever.

Will our worthy ministers now learn, at last, that they were elected to look after health, education, law and order and other matters that truly engage voters' hearts? They were not elected to be building contractors or managers of visitor attractions.