HTTP/2 (i.e. gRPC) is a bidirectional streaming protocol, and you can use the fetch API in JS to use it. The reason gRPC-web exists is because browsers artificially hide some of the headers, which have been part of the HTTP standard since the beginning. If that was fixed, gRPC would just be plain XHR or fetch requests and gRPC-web would go away.
I don't explicitly mention it in the post, but no browser has support for fetch request streaming yet, so true bidirectionality would not be possible even if you had control over the headers. This will come eventually, and then grpc-web will have proper bi-di streaming support. It is doubtful whether it will actually have access to raw HTTP/2 frames which would be required for the gRPC HTTP/2 protocol.
Also, is head of line blocking really that big of a problem for RPC? It only seems to really be a big deal if your latency tolerance is really tight.