I think you're correct to observe that owning an appropriate home can be a good financial decision, but that doesn't make a house an income producing asset (unless you're renting out rooms or have a crystal meth lab in your basement).
In your example of the two persons with a net worth of $1m, one of whom owns a $250k house and one who rents, both have the same net worth and there is no immediate advantage for the homeowner. If there were, the renter could simply take $250K of his money and buy a house. Similarly, the homeowner can sell his house and choose to rent.
Historically, houses haven't been a great investment, certainly not as good as the stock market. One can argue that it is better to rent and use your increased cashflow to invest. That said, I think homeownership is still a good idea. First, it stabilizes your housing cost. Rent goes up unpredictably, and the costs of homeownership are more stable, although you have to be prepared for maintenance and repairs and property taxes. Another good reason to own a home is that it helps diversify your assets. When the stock market tanks, it will be nice to know that you don't have to draw on a shrinking asset to pay rent every month.
I think another good argument against counting a house as an income producing asset is that many people allow their home equity to mislead them into thinking their financial situations are much better than they are. Or, even worse, they delude themselves into buying far too much house because "it's an investment." When the value of their house goes up (on paper) they convince themselves that they should take out some of their "profit" in the form of a home equity loan. (This isn't too relevant on this forum, but I) saw a lot of people get burned this way when the housing bubble burst.)
Person B in your example, with a $250K home and $750 investable has about 25% of his net worth tied up in his house and is probably in pretty good financial shape. Most Americans have a much smaller net worth and a far greater percentage of that net worth in their homes. I think the median net worth of Americans approaching retirement age is under $200K and probably 70 or 80 percent of that is in their homes.
Let's add two more examples to Persons A and B: Person C who has a liquid/invested net worth of $250K and a $750K home. He is also worth $1M on paper, and I would argue that he is in far worse shape than person A or B, particularly if he is retired and has to live off investment income. And how about Person D who has a liquid/invested net worth of 130K, and $870k worth of equity in a 1.2 million dollar home and a $300K salary. He has a net worth of $1M and probably looks richer than persons A,B, and C, but is in real trouble if he plans on retiring anytime soon.
I think you are on the right track to think about the financial value of a house as its rental replacement, but only up to the point where the rental cost is still in some sort of balance with net worth. Ironically, despite all my arguments above, I think homeownership is actually a big positive for most people. I think that most people overspend on their housing. But paying a mortgage is enforced savings/investment in a (mostly) appreciating asset. If they didn't overspend on their housing, they probably wouldn't invest the money, they'd just lease a fancier car and buy more junk. If they didn't have 80 percent of their net worth in their homes at retirement, their net worth would probably just be 80 percent lower.
I think a house can be an important part of your financial plan, but it should generally be viewed as an expense that ties up your money in exchange for paying rent, unless you plan on downsizing or moving to a lower cost of living area. I do think it should be considered part of net worth, though.