A person's "estate" is a vague term to me... if a person does not have much in his "estate," will an insurance policy cover funeral and other expenses?

Don't quite understand your question. Are you asking if the proceeds of the decedent's life policy can be used to pay funeral expenses? That would depend on the beneficiary. Or are you asking if provisions can be made by the policy holder for a set-aside to cover funeral expenses, or are you asking if a funeral home will accept assignment of a life policy for their charges?

An excellent and cost free resource for information on this topic is a funeral home. They are always anxious to assist in such matters and understand all of the steps and procedures, whether pre-or postmortem.

Back in the olden days, new parents would buy insurance policies as "burial insurance" (naming themselves as beneficiaries) in case the child would die.

If someone named me as beneficiary, when he died, I would be hard-pressed to give up some of the insurance money to bury him.

Whoever is named as beneficiary would need to be told by the person who bought the policy and named him that burying the deceased is one of the requirements.