Basically all true, however a home is an investment in the sense over time that it is your cheapest source of money.
An investment is an allocation of resources to productive as opposed to exhausted spending. That a house builds 'equity' is a side effect of nominal price inflation, not a productive use of resources. If inflation shunts into other channels it causes prices there to rise too, and if that becomes long term you may as well call those goods 'investments' too, even if they're waffles. Afterall, if you buy a waffle for a buck and sell it for two, you've made some money haven't you? It only means anything to people who ignore the decreased buying power of the money they're getting.
There is nothing about a house that makes it an investment except the fact that it's nominal price tends to increase and so people see it as a source of money, as you say, and so the government set up a legal fiction to make interest payments in your stead so as to make it seem like you're turning a profit. So you cash in on the nominal price gains and dissapate those on more exhausted spending and are now further in debt than you were before. That, in and of itself, belies the idea of a house being an investment. Monetizing a nominal gain in an asset price spiral, no matter how prolonged, and using the proceeds to fund more debt based spending is a destruction of wealth, not a creation of wealth.
If you truly think a house is an investment capable of creating wealth ask anyone who is now first, second and third mortgaged to the ears beyond the nominal value of their home how wealthy they feel, especially after the bank takes their house, car, SUV, ATV, camper, snowmobile, and dips into their savings - if they have any - to cover the debt that now reflects how massively beyond their means they were truly living. And the people who sold them all that crap, living well before no doubt, will be short a sh*tload of debt funded buyers, production will slow, capital and labor will devalue and be liquidated. And thanks to a wonderful Fed intent on correcting the effects of policy that was too loose for too long with more loose policy, nominal prices likely won't sink except in the most inflated markets as they try to manufacture another bubble so people can keep buying instead of saving, which is what they really should do. So people won't even have a strong dollar to comfort them, something they would otherwise enjoy.
I any case, the test for buying any consumer good is do you want it and can you afford it? No one is *****ing because their computer costs half as much now as it did when they bought it, because there's no bubble in the computer market tricking them into buying it for the purposes of selling it later, or using nominal price gains to finance debt to cover more spending, which, when the inevitable collpase in the price does occur, get exposed as exceedingly bad decisions. That's all that's happening here. That said, so long as you don't want to play Vegas Casino with your life, now could be the perfect time to buy a house before capital up the market gets liquidated and inventory starts declining and demand plus inflation, often times funded by it, starts jacking prices up again. Esepcially if the dollar falls more broadly against other currencies which it just might.