I think his point is that tenants have had no leverage to drive down prices, but now that the landlord will be paying they will pressure brokers to lower their commission.
Indeed. Though it's not that simple.
When the tenant pays the fee, the broker tries hardest to get the highest possible rent, a delay of a month or two doesn't bother him that much, he just wants the highest possible fee, which is directly related to the requested rent. And in many cases the tax pain is mitigated by the fact that the renter doesn't issue a 1099, so a lot of the income might go underreported. The same renter won't come back to the broker in a month or two (and possibly for years) looking for another apartment.
But when the landlord pays the fee, while it will be a cost of doing business and factored into the rent, a landlord might have lots of inventory, and might care more about losing a month or two of rent than getting a higher monthly rent. The landlord might also negotiate with the broker a lower fee in return for higher volume or other sweeteners such as exclusivity etc.