I made the Checkbot SEO extension (https://www.checkbot.io/). Common HTML/CSS errors are fairly minor and aren't going to impact ranking, but are usually worth fixing to prevent potential problems.

Really serious HTML/CSS errors could get in the way of users and search bots from viewing and understanding a page though. For example, broken HTML might lead to broken links, broken heading tags, broken meta tags and broken images, which could have some impact on ranking.

So Checkbot isn't saying you can't rank well if you fail HTML validation (and it's very, very unlikely to be the issue in the article), but it's probably a good idea to fix it. It's really up to you to look at the errors found and decide if it's worth prioritising a fix, like you would with any other linting tool.