Execute snippet test on a private snippet instead of public

What does this MR do?

Public snippets can trigger recaptcha, which causes automated tests to fail: gitlab-org/quality/staging#49 (closed)

So we'll try with a private snippet instead. Private snippets are shouldn't be flagged as spam

Private snippets can't be embedded, so remove that check. Also unquarantine the test.

Closes gitlab-org/quality/staging#49 (closed)

Does this MR meet the acceptance criteria?

Conformity

• [-] Changelog entry
• [-] Documentation created/updated or follow-up review issue created
• Code review guidelines
• Merge request performance guidelines
• Style guides
• [-] Database guides
• [-] Separation of EE specific content