"cash-out mortgage"...

...purely out of curiosity, in the US does it make any difference for tax purposes whether the mortgage was taken out at the time of property purchase, or later?

Canada is a bit unusual.
Mortgage interest is not in general tax deductible.
But that's because the deductibility of ANY interest is not a function of what security was used (a house, say), but rather what the loan proceeds were used for (buying a house, versus some other purpose).

Most mortgages are used for purchase of the property which was used as collateral, which is not "valid" purpose of funds for tax deduction, so most mortgage interest isn't deductible.
But if you pay off that mortgage, then take out an identical new mortgage and use the proceeds for investment, the interest is deductible thereafter because the loan proceeds were for a business purpose rather than to purchase a dwelling.
"Interest paid on money borrowed to earn investment income" is a valid deduction, no matter what the collateral was.
This is not nearly as widely known a "loophole" as you would expect.
Traditionally there were LOTS of people who would be much better off selling their portfolio, paying off their mortgage, getting the mortgage again, then repurchasing their portfolio.

I even did this half-and-half once: I switched my mortgage to a HELOC, paid off half, ran it back up again buying investments, and was able to deduct half the interest thereafter.

(this summary used to be right, but maybe they have changed the tax rules recently...I haven't followed the details there lately)

Jim