[Serverless Search Plugin] Migrate authc.getCurrentUser usage to coreStart.security

[tsullivan]

Part of https://github.com/elastic/kibana/issues/186574

Summary

This PR migrates the method to access a Serverless Search view model field, which consumes authc.getCurrentUser, to use coreStart.security.

Background: This PR serves as an example of a plugin migrating away from depending on the Security plugin, which is a high priority effort for the last release before 9.0.

Checklist

Delete any items that are not applicable to this PR.

• [ ] Unit or functional tests were updated or added to match the most common scenarios