The wrong reason
Do not choose headless because the current site looks dated. A strong platform-native rebuild can still be the better business decision.
The right reason
Use headless when the public storefront needs a different model: richer editorial pages, strict speed and SEO control, better product presentation, or cleaner long-term maintenance.
The test
Before building, name what each system owns: storefront, CMS, products, checkout, customer state, schema, analytics, Cloudflare, deployment, and support.