This is extremely simplistic. Housing is relatively speaking cheap in places where people don't want to live. In places where people do want to live, the cost of land and the cost of heightened demand of construction tends to dwarf planning related costs.

Near me, in a London suburb, there are at least half a dozen ~40+ story residential buildings that have approval where no construction is currently happening because the market conditions are not there to make it worthwhile vs. simply waiting a couple of years, because scarcity in house supply keeps driving up prices to the extent that it's not always clear that building now is the right choice, especially not when there are other buildings going up - it's easy to just sit and wait until there's less competition and start construction then. If planning was all that was stopping them, we'd have plenty of these buildings going up right now.

Importantly, because these developers have a relatively safe bet in terms of returns once they eventually construct, they are able to find investors to finance assembling and sitting on relatively large tracts of land or low density housing, and thus participate in keeping the supply lower, and thus contribute to the house price growth.

There certainly are places where planning is the main thing driving prices up, including parts of London, but the point is that just addressing that tends to be insufficient to address this problem, as cost of increasing density goes through the roof very quickly, and so in hotspots where land is expensive you tend to end up with expensive housing anyway, unless you have governments prepared to force the issue and saturate the market to take out enough demand to drive the price of land down.

If you want to address housing prices in a less heavy handed manner, you need to address the centralization of jobs, which tends to mean addressing transport. A pet peeve of mine for London, which applies equally to a lot of big cities, is that addressing the housing and congestion issues in the centre is a net negative overall:

It costs a fortune to drive up the density and transport capacity in an already dense area, and it just attracts more demand until things are just as bad and the next "solution" is far more expensive. The more cost-effective solution tends to be actual planning on a larger scale to make bypassing the expensive/congested region easier and more attractive.

E.g. in the case of London there are a number of towns that feed a huge amount of people into London. Most of these towns would find it a lot easier to subsume additional housing density at far lower costs. What they lack is the ease of transport from a large enough base to be more attractive places for companies to locate. What's required to kickstart that is largely an initial willingness to invest in upgrading transport links to other places than London. Ironically many of these towns are "new towns" [1] that were in many cases constructed pretty much entirely from scratch to reduce strain on London (and other cities) after the war. And they achieved that, but instead of building on that and also decentralising transport to encourage these towns to grow as hubs in their own right, transport has instead become increasing focused on making commuting in to London faster, so even though many of these towns have their own small commuter belts they are to far too great an extent feeding the problems in London.

You don't solve that by letting planning loose, because a lot of these investments are beneficial to wider society but not necessarily profitable individually unless you can take into account how they affect overall economic growth and tax receipts.

[1] https://en.wikipedia.org/wiki/New_towns_in_the_United_Kingdo...