Do I have to have a medical condition (or be caring for a family member who has one) because of a major disaster or emergency in order to apply to receive leave under ELTP?

No. The ELTP covers more than just medical conditions.  Perhaps you need time to file insurance claims for lost property, or to assist a family member relocate from a major disaster or emergency area.  In order to receive leave under ELTP, you need only to demonstrate “severe hardship resulting in required absence from work.”