I think that the tenant does have a plausible argument that the language, "tenants are responsible to maintain their own renter's insurance", is ambiguous -- it could be interpreted as requiring that tenants carry insurance, but could also be construed as a statement that the landlord does not provide insurance for the tenants' property or personal liability. It is possible that there is other language in the lease that would render the language clear.
The big issue is that, sooner or later, the landlord is going to require proof of insurance. It's also a good idea for tenants to purchase renters insurance. So the tenant can argue that the lease is ambiguous and perhaps hold off the issue until the next time the lease is renewed -- or for a rental period (or at most two, depending on state law) if this is a month-to-month tenancy. Ultimately, if the tenant does not want to purchase renters insurance by the time the lease is up, the renter needs to think about moving.
I doubt that a court would find the failure to purchase renters insurance to be a material breach of the lease, particularly if the clause has not previously been enforced; but even if not, it's not the sort of breach that would typically support an eviction, and it is also the type of issue that a court would typically allow to be corrected if it found that eviction was otherwise warranted.

