An injured tenant could just sue you personally for negligence in your management of the property.  

I spend over a $1000/month on premiums and last week they would not cover a tree that fell on a tenants car because the tree did not look dead so I was not 'negligent.'

Do you still think it's a big liability even if I disclose and am not negligent regarding repairs? 

That's a common structure; however it can be Overkill. you are fairly protected by negligence since you can't actively handle most affairs of the property.

If LLC A has $2,000,000 in assets and LLC B only has $1,000,000 but LLC B gets in trouble -- lawsuit from a tenant alleging negligence on your part led to her falling down an improperly maintained stairs with a baby, and this led to her and the baby all sorts of injuries, cracked skull etc..If the court enters a judgement of say $1,500,000 against LLC B, the most you can loose in whole is the $1,000,000 of assets in LLC B.

Once someone at the bank realizes they have lost the property they might go after the surplus without realizing their own negligence was the reason the property was lost in the first case.We have taken numerous properties from banks that were asleep at the wheel.

Really, the extra cost to doing each individually is negligible.

Bottom line: the difference is negligible in this context.

Wouldn't it be their liability, at least if it was due to some negligence on their part, if you had losses related to their fire?

The cost of the loan is usually negligible if anything.