Should I give a recorded statement to the insurance company?

Two different situations.

The other driver's insurer. You're not obligated, and there's little upside. The recording is permanent, it's taken before you know the extent of your injuries, and questions are structured around specific issues — prior injuries, gaps, your activities since.

Your own insurer. Your policy generally requires cooperation, which may include a statement, particularly for PIP or uninsured motorist claims. Refusing outright can jeopardize coverage. But you can ask to schedule it, ask what it will cover, and have counsel present.

What makes recorded statements dangerous even when you're truthful. Memory is imprecise. "I think I was going about 40" becomes a fixed admission. "I feel okay" on day three becomes evidence you weren't injured. Estimates get treated as measurements.

If you give one: answer only what's asked, say you don't know when you don't, don't guess at distances or speeds, and don't characterize your injuries before a doctor has.