How is that collective bargaining when your neighbors have different landlords?
All you're suggesting is to stop paying rent, which anybody can do on their own, with the usual consequences of your credit getting ruined and then the landlord serves you an eviction notice and changes the locks.
Consider the economics of what you're suggesting. Your unit costs $800k when it should cost $200k because of artificial scarcity (zoning), but then the landlord has to pay $800k for it and charge rents high enough to pay back that loan. The way you fix that is by getting the cost of the unit back down to where it should be, i.e. building more units to alleviate the scarcity. Demanding that the landlords charge less without reducing what they have to pay for the building doesn't work, because then who is going to sink $800k to build a new rental unit? Who is going to keep renting out their existing units at a loss instead of turning them into condos or similar? Which then makes the scarcity of rental units worse and causes rents to go up.
On top of that, "puts them out of the game" means that there are no apartments to rent. And it wouldn't even work because you'd not only still have the non-landlord homeowners lobbying to suppressing the housing supply, you'd have the banks making sure you can't actually wipe out the landlords since the landlords owe them way too much money for things to happen that way.
What else do you expect "collective bargaining" to do in a market where every tenant has a different landlord? Your landlord isn't going to care if the tenants of 10,000 other landlords threaten to move out. They only care if you threaten to move out, and then to credibly do that what you need is to show them the unit you could move into which is asking for less.
The real problem is that you need the unit you can threaten to move into to first exist, rather than its construction being inhibited by zoning etc.