Is state entitled to any money leftover from my mom’s life insurance policy payout?

No, not exactly. There are some exceptions here. In the state of Ohio, you have a few options. You can designate what is leftover from the life insurance payout to pay off some old debt. You can have it go to the estate. As long as you are the primary beneficiary, these actions are usually left up to you.

Some states, like Ohio, use the example of "probate" to sink their teeth into some of that money.

One way to avoid the possibility of "probate" is to have it play out like it did. She left everything to you. This is a good thing. When it is left to a beneficiary it passes through everything unencumbered. This goes for everything from the estate to the life insurance policy.

Since you have avoided probation, no creditors from the state can attach themselves to the money. There is one exception to be noted here.

Did you or your mother make a deal with any creditors she owed money to? Did you guys sign a contract that you would be responsible for her debt when she is gone? These are two exceptions where it is possible for them to come in and take back some of that money for the state. Let us say for argument's sake that you did not do this. The money is clear. Legally, the state cannot take any money from you.

It is up to you to dictate what you want to be done with the money after the funeral is taken care of. Talk to you lawyer to get a definitive answer on this. The laws are changing every day. States tend to find legal loopholes in order to get their hands on the money. There should not be a problem with the life insurance policy though.