I feel like you reiterated my point, are you disagreeing?

Increased supply (more units built to rent), decreases demand (less people wanting to rent) by increasing competition (more units on the market vying for the same set number of renters), means decreased rent.

I suppose my point of leaving a unit empty or decreasing rent was somewhat reductionist, but still what will happen.

Ultimately the only point, that isn't up for debate is that... More units built = cheaper rent.