Thanks for the find Sello_2k
That allows us to generate a token but sadly, not to use it.
In order to generate a token using this mutation, you must create a private app with both storefront API and admin API scopes. This mutation can then be triggered.
This will yield what appears to be a valid storefront access token.
In my trials, I was able to make requests to the storefront API api/2022-07/graphql.json when using the generated token under the header X-Shopify-Storefront-Access-Token.
Unfortunately, if I switched to using this token in the intended header of Shopify-Storefront-Private-Token I receive a 403 error.
It should be noted the store this is being on trials on is not a Shopify+ store. But that isn’t currently mentioned as a restriction to the use of this token in the documentation.
So it seems we are still stuck.