Hey,
You're welcome - I'm glad you found it helpful! Firstly, a few things to note about the background check. Firstly, most programs only pull it once you're admitted. Secondly, they're not always full criminal profiles. Some do, some don't. It varies. However, the first point here is the most important. The background checks that schools perform are time intensive and outsourced to companies that specialize in this. They verify your employment, salary, recommenders, etc. It's a time intensive process and it makes the most sense for them to do it after they admit you - just to make sure you are who you say you are. At that point they'd have to decided if they're going to rescind your offer and it is highly unlikely (but not impossible) that they would do so.
I don't think it's bad to own something that happened in your past but the optional essay may not be the right place to do that. There might not be any right place. Think of it this way - HOW would you own it in that essay? Would you merely acknowledge it? In which case you're better off just not writing anything at all. Would you provide a reason for it? I doubt you have a very compelling reason for a DUI. Would you say you learned from it? If so, then we run into the same reason it wasn't good to write about in the failure essay.
Overall, I think you're better off just leaving it alone. If they pull your record before admitting you, it's on there and that's a fact of life. If they don't, then it doesn't impact the admissions process and it doesn't matter until a little bit later down the road (if even then).
You're not the first applicant with a DUI and you won't be the last. They're unfortunate and they do highlight a documented lapse in judgment, but they're not the end of the world. It's natural to fixate on the part of your profile you're most worried about. There just has to be careful thought around the messaging and strategy.
Bhavik