I can understand the reluctance to use the “open to work” banner on sites like LinkedIN. Unfortunately, there are a lot of old-school people out there that would look at that being a negative, but ask yourself this: would you really want to work at a company where folks like that are in charge of hiring? We’re no longer in an era where people being laid off is an accurate indicator of performance. Much of it is being at the wrong place at the wrong time.
I don’t have any data on if things like that help or not. Anecdotally, I can tell you that several years ago, an acquaintance was laid off, turned on that banner, and I happened to see if because a mutual contact commented on his post. I immediately submitted him for a role at my company and he was hired. The point is, many folks who you may have worked with years and years ago may remember you, may remember the good work you did, and may also have an inside track on new positions coming in their company and oftentimes, they’ll be incentivized to refer people due to referral bonuses.