This is only my two cents but worth a read. If the bathtubs waste-overflow did not work, then your recovery for damage may be limited due to the a drain's malfunction.
It could have been: the overflow gasket was old and cracked, or a prior plumbing snaking the drain thru the overflow let the gasket fall down and re-installed the overflow cover without the gasket between the tub and overflow pipe. This means water entering the overflow could escape behind the overflow cover. May not have been the tenants fault.
As a licensed plumbing contractor who can explain to the owner, tenant, insurance carrier or renters insurance what was the actual cause of the leak, I OFTEN TAKE A VIDEO AS I TAKE APART A PLUMBING FIXTURE/DRAIN TO DETURMINE THE CAUSE AND FAULT. I have testified for both owner and tenant to determine fault.
In one case involving tremendous water damages leading to MOLD issues, I testified the owner re-rented an apartment without checking to see if the drains were running slow or needed a snaking. The insurance carrier thought this meant they were off the hook. But when the owners attorney asked me how many landlords snake out drains between tenancies, I had to answer, "only landlords who are also plumbers due such a preventive maintenance task, such as myself! The other 99.9% don't".
Also your tub spout may allow 6+ gallons a minute to be released, but your overflow drain may be tiny and limited to only a few gallons a minute and water could rise above the tub level and drain onto the floor.
How did the water escape the tub? From the overflow or over the tub itself?
If you get just the average plumber, he may not be able to prove who's fault it was. Without proof, the renters insurance policy won't pay and your landlord policy may limit coverage.
ALSO YOU HAVE TO READ THE POLCIES. I remember when a policy said it would pay for "water damages", so long as you make a claim -- without stating any issue about "fault". All you have to do was, "MAKE A CLAIM WITHIN THE NEXT 2-3 DAYS". And you were not required to tell the insurance carrier how it happened, who caused it. You could say, "I DON'T KNOW". And that was good enough to get the claim paid. Too many details could cost you money. So depending on your coverage and who caused what will determine your next step.
Good luck! --47.155.xx.xxx