By your definition, a stock that does not pay dividends would not count as an "investment" either because you can only realize capital gains on it (if you get "lucky").
The stock is still generating value. The house is not generating value unless you go out of your way to rent part or all of it.
You could just as easily rent out bicycles or hammers, but no one would go so far out of their way to class a bicycle or a hammer as an investment.
All assets generate a return, some just do so badly (or negative returns). Anything that's not an asset is a consumable (like food).
And how is the rent "income" not offset by an equal rent expense?
yes, in a sense. The fact that you're not getting paid a rent from somebody else doesn't mean there isn't value. You could think of it as opportunity cost - owning a property but not getting rent from it.
> And how is the rent "income" not offset by an equal rent expense?
it is exactly offset-ed if you lived in that property.
negative expected value = gambling
every other money game is a synonym of either of those