Is it better to buy or rent?

True, but there's always the risk that where you like living changes over time and becomes HELL 20-30 years down the road. And I have seen this happen among several of my family members.

When you rent, sure, you can't stay forever, but you have visibility at 2 years down the road (depends on the country where you live, but where I do the owner cannot get you out for the duration of your lease, and that's usually 2 years).

It's not clear to me that, say, $500k of assorted shares + $500k of house is worse than $1M of assorted shares. You've got more in a single asset, which is bad, but you're diversified into more asset classes, which is good.

The picture is a bit different if that $500k is all your assets or even substantially more than all your assets (a common situation for a first-time buyer with a big mortgage). Which suggests you should rent until you've built up substantial investments of other sorts, and then consider buying.

Historically, I think the variability in price of a single house hasn't been any worse than that of, say, a typical index fund. (Disclaimer: I haven't actually checked the figures.)

Of course, all the above treats houses as just another investment. This is a useful perspective, but you're going to be living in your house too, and questions like "can I redesign the bathroom?" and "if there's a flood or an earthquake, am I liable for fixing/replacing everything?" and "am I bothered by the possibility that my landlord might just throw me out one day?" and "can I get up and move somewhere else with minimal hassle?" may be just as important to you as "what is likely to leave me with more valuable assets 20 years from now?".

This is actually sometimes the case, though please double-check your analyses before committing to outliers or "betting the farm". Since it does happen and can work for some people / some places it certainly complicates the simplified dichotomy presented in the OP and this topic.

There is an important oversimplification that these formulas/programs/spreadsheets often make, and I see it happening here. Consider two alternatives where everything is the same except the amount of money used for the mortgage down payment. The formula assume that money not used for the down payment is invested at a certain rate of return. If the mortgage interest rate is greater than the investment return, then the formula always indicate that the largest down payment possible should be made and if the rate of return for investments is assumed to be greater than the mortgage interest rate then the smallest possible down payment should be made.

Its easiest to understand the problem with specific numbers. Assume a home buyer has $120,000 that could be put into a down payment, but that the minimum required is only $20,000. What should the buyer do? Assume that the investment rate of return is 5% and the mortgage rate is 4%. Investing the $100,000 instead of using it for the down payment is essentially borrowing $100,000 at 4% and investing it at 5%. All the formulas/worksheets/programs like this one and even professional investment advisors will end up showing that is better to put down the minimum down payment and investing the $100,000. This ignores the risk between alternatives. Putting the $100,000 into the down payment produces a guaranteed, riskless, return the buyer of 4% per year (by saving him or her the mortgage interest payments on the $100,000). The 5% potential return from investing the money isn't a fair comparison. The comparison needs to be made to a riskless investment (e.g. US Treasury Bills). Currently the riskless rate of return available to investors is approximately 0%. This means that in the current environment the correct alternative is to put the $100,000 extra into the down payment (absent any liquidity concerns).

One may say that they are willing to take some risks to obtain a higher rate of return. Modern Portfolio Theory has its detractors, but as far as home buyers are concerned, its implications are still apropos. Having one component of your overall portfolio earning the equivalent of a riskless 4% (by making the larger mortgage down payment) is likely to produce better aggregate returns (on the home and additional equities etc.) at whatever risk tolerance one designs for their overall finances.

I have no formal training in finance or investing so none of what I've described here should be interpreted as advice, instead it is intended to spur discussion.

It depends a lot on where you live in the US. In big cities, I'm sure it's definitely cheaper to rent. In more rural areas it's often less obvious which is the cheaper option and it really comes down to preference. And sometimes in rural areas it's cheaper to just own, depending on the cost of the houe.