The performance argument is weaker than it was
Online Store 2.0 themes on Shopify’s CDN already deliver good Core Web Vitals when the app stack is controlled. Most slow Shopify stores are slow because of apps and images. Going headless without fixing those carries the same problems into a more expensive stack.
Hydrogen on Oxygen does give you streaming server rendering and tight control over what ships to the browser. That is real. It matters most on very large catalogues, heavy filtering and markets with genuinely poor connectivity.
Count the total cost, including the team
A headless build typically runs several times the cost of a themed build, commonly quoted in a three to five times range, and the gap widens over time. Every section your marketing team could previously edit in the theme editor becomes a developer ticket unless you build a content layer.
Add a CMS such as Sanity or Contentful and you have a second subscription, a second set of permissions and a second place for content to go stale. Budget for ongoing developer availability rather than a one off project fee.
What actually breaks when you go headless
Theme app extensions stop working, so reviews, subscriptions, loyalty and upsell apps need API integrations or replacements. Check every app in your stack for a Storefront API path before committing. This is where headless projects overrun.
Checkout stays with Shopify either way, which is good news. You also take on responsibility for SEO fundamentals the theme handled for you, including canonicals, sitemaps, structured data and pagination. Plan those into the build.