Why are new backlinks missing from my Google Search Console Links report?

That can be completely normal. Google says the Links report is a sample of links it has found over time, not a complete or live list of every backlink. It also combines duplicates, groups your URLs by canonical version, and limits the report tables, so a real link can be absent or appear under a different canonical page. For a better look at recent discoveries, open Links > Export and download Latest links; Google says that export can include up to 100,000 rows and is sorted by discovery date. Check that the linking page is publicly reachable and that the link points where you think it does, but don't assume a missing report entry means the backlink is broken or needs action.