Description
Following up on the discussion on twitter, the problem is that the proposal replicates functionality that one can create using already-standardized web technologies. As you argue in the README, on their own, the programming primitives are not straight forward to use (XHR, CORS, etc.); however, once a library is written to implement the API you are proposing, and the search providers agree to enable CORS for the JSON file, that becomes essentially a solved problem (without any additional help from a UA).
I would strongly urge you to build a JS library and consensus amongst interested parties. That will still allow for standardization of things like the JSON file location, while allowing the API to grow and adapt organically to the needs of search providers. It will also allow you to build more flexibility into, for instance, pagination and specific search filtering parameters.
For more rationale on why browser vendors prefer to only work on low-level APIs, please see: https://extensiblewebmanifesto.org/