The two big downsides I see are that:
First, it needs to be filled right before whatever disaster causes you to lose city/well water.
Second, once it's filled, you can't really move it. So, you also lose the ability to use your tub for anything else (like bathing with the water in your water BOB).
While I guess it might be worth having for someone that lives in an apartment and can't normally store much water, but if I have a house with some storage space I'd rather store water another way. Better to have plenty of water stored in advance than having to worry about filling your tub right before an emergency hits.