Although primarily for contents, renter's insurance also provides liability coverage for injuries and other damages visitors sustain while on the property, as well as damage the renter inadvertently causes to the building. The liability aspect is important to the landlord because a litigious, injured visitor may go after the landlord in addition to the renter. Also, if the renter has insurance and suffers losses, then he is probably more likely to simply file his insurance claim rather than (wrongly) sue the landlord.
That is all true. But you must understand that liability is either present or not and has nothing to do with whether you have insurance.
Let's say that you are a renter and you leave a hose across your walk. Someone trips over it and is injured. You and the landlord may both be liable - you for your action and the landlord because it occurred on their property. The injured person would always sue both of you and if he won he would recover based on the proportionate liability of both you and the landlord. Whether you are insured is not relevant until it comes time to pay up.
Long ago when I rented we were broken into. We had renter's insurance and we filed a claim. Our insurance company paid us immediately. But, then they went after the landlord for the full damages PLUS administrative fees and other costs. Since the burglars broke in by breaking the door lock and the landlord is required to provide secure locks, they were determined to be liable. Clearly it could be different in other states but it seems bizarre to me that a landlord would want to require insurance. If we had not had renters insurance we would not have filed a claim and the landlord would not have paid out anything. By requiring insurance a landlord is essentially inviting the aggressive action of the renter's insurance company against them when something happens.
And, the fact that the coverage might extend to damages caused by the renter would also scare the crap out of me as a landlord. It would put me in the position of dealing with an insurance company bureaucracy rather than just dealing directly with a tenant over damages.
On the other hand, I happen to rent a place out now. I carry insurance for my liability and it is dirt cheap, less than half of what that quote for renter's insurance is.
I do think renter's insurance is a good idea if you have valuables to insure or if you have a high enough net worth to be concerned with personal liability. But as a landlord it does not seem like I have any interest in whether or not tenants cover themselves. If anything I think I would prefer that they NOT have renter's insurance.