“A marathon and not a sprint”: that was how London mayor Sadiq Khan described fixing London’s housing crisis as he launched his draft housing strategy last week. But even “marathon” seems an understatement when you take a quick jog through the numbers.

Total housing completions in London are expected to peak at around 46,500 this year, according to research published by Savills in April. While this is ahead of the minimum target of 42,000 homes set by the London Plan, it is still far short of demand, which analysts put at between 50,000 and 64,000 a year.

Meanwhile, the UK’s historic obsession with home ownership has led to a woefully short supply of decent rental homes, fuelling an affordability crisis that has left Londoners needing an average deposit of £90,000, which many – paying out more than half their income in rent – have no chance of saving. The biggest casualties of this state of affairs, though, are the estimated one in 50 Londoners who have no home at all.

The factors behind London’s housing crisis are complex and interlinked. The historic focus on homes for sale is far from unique to London; but this dynamic, combined with the geographic imbalance of the economy, has ensured that in the capital demand far exceeds supply, and prices are beyond the reach of average wages. Meanwhile, the high prevalence of brownfield sites has contributed to high building costs, and the cost of meeting planning obligations has often priced new entrants, particularly those willing to develop small sites, out of the market.

However, there are two huge barriers to solving London’s housing crisis, which the mayor has no control over

Khan’s housing strategy is a laudable step towards tackling this many-headed hydra of issues. His proposal to reduce upfront Community Infrastructure Levy payments on small sites should help encourage SMEs into the market. Meanwhile, he has already gone further than his predecessor, Boris Johnson, to incentivise the development of private rented accommodation, and his strategy puts forward further support for this.

The mayor’s election promise to require 50% affordable homes on sites has already been watered down as he negotiates with developers over the viability of the provision, particularly on sites acquired before his election. But as Khan tries to move the sector towards that 50% goal, initiatives such as his push on private rented will start to alleviate the affordability issue.

However, there are two huge barriers to solving London’s housing crisis, which the mayor has no control over. Unless they are addressed by central government, his admirable efforts can be little more than tinkering with the pieces of a broken system.

The first, and the most glaring, is public funding. Historically, the UK, and London, has only built the volume of homes that are needed now when there were huge public housebuilding programmes sitting alongside private construction. The mayor, largely reliant on funding allocated by government, with a modest top-up from GLA funds, is hamstrung on delivery without more financial support.

New building methods, like modular construction, may well open the door to creating more homes, faster, for less – but for firms to make the sizeable investment needed to get that technology up and running at scale, they need to see evidence of a stable pipeline – something the public sector is uniquely placed to provide.

If the housing crisis – in London and elsewhere – is really the national priority it should be, that warning must not be ignored

The second major barrier beyond the mayor’s control is the impact of Brexit on construction costs; in particular, the cost and availability of labour.

The housing strategy nods warmly towards helping construction address its skills shortage, particularly with the launch of a construction academy. But initiatives like this are long-term projects, and London – where one in four workers is from elsewhere in the EU – is staring down the barrel of a short-term skills void.

The suggestion, apparent in a Home Office document leaked to the Guardian last week, that the UK is preparing a hard-line stance on immigration – under which free movement of labour will end immediately after Brexit, and lower-skilled workers will be allowed UK residency for a maximum of two years – makes this all the more pressing.

In a rare act of collaboration, nine major built environment industry bodies have jointly written to Brexit secretary David Davis, warning that if the government’s immigration system after Brexit fails to take account of construction’s specific circumstances with a transitional deal, the “strength of our economy and the delivery of the government’s ambitious targets for new housing and infrastructure will be undermined”.

If the housing crisis – in London and elsewhere – is really the national priority it should be, that warning must not be ignored.