I've never seen an extended-stay hotel in a neighborhood that I would visit for leisure. They're usually in pretty ugly, utilitarian spots - off a cloverleaf or behind a strip mall - serving business travelers.
That's probably true in the US; much less so in Asia or places with good public transit.
There's a Hyatt House I pass by sometimes right next to the Shibuya crossing; though I'm guessing that one definitely does not have the price advantage.
I don't know how you would easily book them if you're not in the system, but what you're describing is a timeshare rental - suite rooms in resort-like environments.
