[ML] Removing error object check in inference API

[jonathan-buttner]

This PR refactors the error handling logic within the inference API to remove some error handling code that is no longer needed after a previous refactoring PR.

Prior to this PR: #123076 we needed to check for the presence of an error object in the response from a 3rd party service. We needed to do this because some services would return a 200 and not label the message as an error within the SSE. So we would check if the body of the response looked like an error and if so we'd parse it as an error and return a failure.

After the retrying work for streaming we moved the structure check within the streaming parsing logic and no longer need to do it within the BaseResponseHandler.

[elasticsearchmachine]

Pinging @elastic/ml-core (Team:ML)