The reason for weekly rent in the game is to experience the landlord's content and the consequences of not paying rent relatively quickly. Imagine having to wait an entire in-game month before you see the next landlord scene, and if you're trying to corrupt Angel as fast as possible, it would take several months... and when compared to how quickly the things escalate in the rest of the places, like the bar, or the subway, or the night attacks, the rent/landlord content would seem underwhelming or incredibly dull.
Weekly rent is only an absurd phenomenon in Australia, New Zealand, and some places in the US. It doesn't necessarily apply to the game; it's just one of those adult game logic cliches, and unfortunately, it won't go away anytime soon.