According to the Modigliani–Miller theorem, the mix of debt (size of mortgage) and equity (down payment) is only irrelevant in a world without taxes. However, mortgage interest is tax deductible, while rent is not, so their result doesn't apply here.
The flat down payment chart is a coincidence, based on the default values for interest and inflation rates, growth rates and expected returns.
I don’t think it’s a coincidence — those default values reflect the current market, and it’s a reflection of the market being fairly efficient. For example, if investment return rates were consistently higher, you could expect mortgage rates to rise as well. Taxes introduce some inefficiency, but I think the broader point still holds.
The flat down payment chart is a coincidence, based on the default values for interest and inflation rates, growth rates and expected returns.