I've had the same problem. My solution was to use the storefront API (Headless channel).
The getting started page for Hydrogen does say that the Hydrogen Channel on your store is a requirement, however this can't be used on a development store (which was true in my case).
Download the Headless channel, here are the getting started docs, follow up to step 2 for the storefront API. This will give you these env variables:
PUBLIC_STOREFRONT_API_TOKEN
PRIVATE_STOREFRONT_API_TOKEN
Instead of step 3 in the docs, go update your .env file. Change the existing PUBLIC_STORE_DOMAIN variable with your store url i.e. {store_name}.myshopify.com. You can go through the same process to add the customer account API, haven't done that myself yet.
Use npm run dev and the store should be using your product information.