It's worth tidying up, but H-Tags aren't as powerful as they used to be (especially h2 and lower). Best practices are always worth following, but when this happened to us I know I stressed it more than I should have.

Titles, meta descriptions, and body content are all more worthwhile projects you can control that will impact rankings more, but for peace of mind I'd have your dev team understand that H-tags aren't to be used as an easy way to style some text.

I agree with you that those nav links shouldn't be h2 tags, but don't worry about whether it might "completely confuse google."  Even from 2009, google was great about processing pages with less-than-stellar syntax. Here's a youtube of a google webmaster (matt cutts) explaining that a similar issue isn't confusing to google. https://www.youtube.com/watch?v=iR5itZlq8sk

We had a similar issue regarding H-tags on our site that we thought was more concerning than it was. It might be a bigger problem if they were all H1s, but I wouldn't stress it as a website breaking issue that must be fixed immediately (though I would, like you, prefer to correct it).

I agree with you that those nav links shouldn't be h2 tags, but don't worry about whether it might "completely confuse google."  Even from 2009, google was great about processing pages with less-than-stellar syntax. Here's a youtube of a google webmaster (matt cutts) explaining that a similar issue isn't confusing to google. https://www.youtube.com/watch?v=iR5itZlq8sk

We had a similar issue regarding H-tags on our site that we thought was more concerning than it was. It might be a bigger problem if they were all H1s, but I wouldn't stress it as a website breaking issue that must be fixed immediately (though I would, like you, prefer to correct it).