web_search is not a toolset provided by github-mcp-server. The complete list of toolsets that this server exposes (as documented in README — Available Toolsets) is:

• Local server: actions, code_security, context, copilot, dependabot, discussions, gists, git, issues, labels, notifications, orgs, projects, pull_requests, repos, secret_protection, security_advisories, stargazers, users
• Additional remote-only toolsets: copilot, copilot_spaces, github_support_docs_search

There is no web_search toolset on either path, which is why copilot --add-github-mcp-toolset web_search doesn't expose it — there's nothing to enable on the GitHub MCP server side. That setting flag just maps to the toolsets above.

The "Web Search" feature you toggled at https://github.com/settings/copilot/features is a separate Copilot platform feature; it's surfaced through whichever Copilot client integrates it, not via this MCP server. As of today the Copilot CLI does not seem to expose Web Search as an MCP tool — the toggle controls the in-product chat experience.

Your org memberships shouldn't matter here because (a) you're on a personal Pro subscription and (b) even if a stricter org policy applied, the result wouldn't be a missing toolset — it'd be a permission denial on a tool that does exist.

If you have a link to where copilot-cli/issues/57 claims web_search is provided through github-mcp-server, it would help to verify whether the doc is outdated or whether a separate integration is being planned. Right now, based on this repo's source, there is no such tool to discover or enable.