Instead, they miss the cloth or aluminum wiring and poor circuit allocation and instead show you with their fancy plug that "polarity is reversed"; or the always arbitrary "the roof likely only has 5 more years left max" because they see a 3-tab shingle from the ground and completely miss hail/wind damage on a newer architectural shingle citing its "lifetime warranty".

The disposal in the kitchen is damaged and doesn't work.

Insurance companies, and this includes umbrella policies, will deny or challenge your claim especially if the damage / claim is high and based on negligence or any fault of your's, READ your small print and don't think you are so good at landlording that accidents won't happen.

The security deposit would also be transferred to the new owner (although you are bound by the original lease to return it after damages are paid when they move out) and a lease supersedes a sale so the current lease stays in place, you don't need to sign a new one. 

Can I use some of the tenant's damage deposit to pay what my insurance does not pay? 

A good operator will have insurance that protects themselves and you as the landlord from damages and liability.Is your property in the OC?

I’ve been with the same insurance company for 5 years and never had a claim and likely won’t make a claim unless I sustain a large amount of damage somewhere.

The vast amount of damage is normal wear and tear.

In general, when the landlord's property causes damage to tenant's property, the landlord is liable, unless there is proof that the cause of the water leak was the tenant (unlikely I presume).

@Malick NgomGet an attorney to send a certified letter requiring signature and include the satisfactionIn most states a lender can be held liable for damages for not recording the satisfaction