Also, multiple homes mean multiple maintenance bills, and also the potential for bad tenants wrecking the place. I had a rental property in the past and had to spend multiple thousand dollars staging it for sale because, renters generally DGAF about maintenance, and property managers get lazy and do the bare minimum to get it rented.
Plus “moving to a less desirable area” doesn’t really dovetail with most people’s assumption of “increased personal wealth”. It is sometimes a good option for retiring somewhere cheap, which absolutely reflects a real increase in net worth. But it’s far from liquid worth and requires steep compromises not everyone is a fan of.
If I can be approved for a mortgage or have cash on hand, and have someone agree to sell me a house, and I sign a contract with them to purchase said house on such-and-such a date, and I fulfill my end of the deal and don't default on the contract, then yes, I am entitled to ownership of a single house. Because I bought it fair and square, and it's now legally mine. Just like if I were to stop making payments on said house, the bank would then seize it, and then I wouldn't be entitled to own that house any more, because I defaulted. This is not rocket science.
This is true only on first approximation. On second approximation rent prices are positively correlated with property prices. This means that rent_amount / mortgage >>> 1 and if you rent the place out you can essentially get passive income. You are correct that you still need to live somewhere and that is going to come at a cost which will eat up the passive income, but effective rent will still be lower.
Another option is to realize that lower mortgage compared to equivalent rent means that extra cash is freed up towards other things. This can either be extra investments every month, experiences such as traveling the world, or even a lower paying job with a lot less stress.