is there a sensible way to check backlinks in google search console?

Links report, left sidebar right at the bottom. "Top linking sites" is the useful table, click into it and you can see which of your pages each site links to, then export. The bit that catches people: it's a sample, not every link, and it's slow to update. A link you got last week might not show for a while. Nothing wrong with your setup if numbers look low.