What the flag is, and what it is not
Recruiters moving off LinkedIn Recruiter ask about this filter before anything else, so here is the plain answer. There is an open-to-work search, it works, and it is not the one you had.
The filter you had includes a private setting, which members share only with LinkedIn's own recruiter product. Nobody outside LinkedIn can see that part. What exists outside is the public version: the badge a person chose to show everyone, as recorded by one people database, along with the things people write when they are looking. "Seeking my next role" in a headline. A contract end date. A post saying they are free from next month.
That is a smaller pool than the one you had, and a more deliberate one. Someone who shows the badge publicly has decided their employer can know. For a contract role, an urgent backfill or a volume desk, those are the first people to call.
What you get back
A CSV ranked against the spec, one row per person: current title and employer, dated history, the score and its reason, and a column naming the signal they showed, whether badge, profile wording, post or a recent end date. Personal email where one was found, LinkedIn profile for everyone. Nobody has been contacted.
Variations worth knowing
Tenure instead of the flag. The same database filters on time with the current employer. People a couple of years into a role who show no flag are often more movable than flagged people everyone is calling. Ask for both lists and compare them.
Contract desks. Add the contractor wording to the read step: "available from", "contract ending", "immediately available". For contractors the profile text is often a better signal than the badge.
Widen to the passive market. When the flagged pool is thin, the next step is a full talent map, where open-to-work is one angle among five. For one role and a standard search, start from the job spec.
Where this goes wrong
Searching the flag alone. Without titles and a location it matches a pool the size of a country. The play always pairs it with the spec, and you should keep it that way if you rewrite the prompt.
Reading no flag as not interested. Most people who would move never show a badge. This play finds the ones who raised a hand. It says nothing about everyone else.
Trusting a stale badge. A database records what it saw on its last visit. The work history check catches people who have started a job since. It cannot catch someone who accepted an offer yesterday.
Filtering on anything protected. Availability is a fair criterion. Age, family circumstances and the reason someone is out of work are not, and the play does not score on them.