See https://www.domainsherpa.com/domain-name-dictionary/life-cycle-of-a-typical-gtld-domain-name/. They start it in auction listings after it expires and before pending delete. I don’t know the exact dates, but if you monitor it you’ll see it go from auction, to closeout, to drop. You’ll need an auctions.godaddy.com account (I believe $5 per year) to bid and buy.
Here's an old post on search engine roundtable that claims google's policy is to discount previous backlink juice when a domain changes ownership. I'm not convinced whether this is actually true or something Google says to discourage excessive domain buying / 301 redirecting for SEO benefit. The comments above seem to give varying opinions on this matter. Would be great to get to the bottom of that one!
Also, I currently have two other domains that are registered with Squarespace, and I have their annual payments set up automatically so I never have to manually pay (and so nobody else gets that window of opportunity to take the domain away from me). Would it be possible that the owner of this domain might just have a similar system in place and that I might be wasting my time?
We do not only have Expired Domains, but you can also find lists of Deleted Domains for a lot of TLDs. Deleted Domains or Dropped Domains are available for registration and can be picked up for just the normal regfee at your preferred domain registrar! All Domains have the typical SEO relevant data, like Number of Backlinks, Archive.org Birth Date and a lot more.