Over time, they will probably wonder: "hey, why is this person investing so much time into helping me?" That may be enough for them to get the picture.
Also, it sounds like you're worried about conflict of interest since this person doesn't report to you. Would it make sense to approach their manager with something like: "I've noticed EmployeeX doing X at work. Have you noticed that too?" That helps see if you're in alignment with the employee's manager. Maybe the other manager has reasons for encouraging or tolerating that behavior. That's important for you to know before you offer to change it.
If you are in alignment then something like "It reminds me of a situation I was in earlier in my career. Do you mind if I ask EmployeeX if they'd be open to some coaching around the issue?" That way you get everyone on board.
Transparency and honesty will usually help diminish people's fear of "interference". And if the manager or employee aren't willing to accept your help or see things as you do, then it's better to accept that and focus on your role and responsibilities.
No matter how nicely you put it, you're going to come across as "I know more than you, I noticed this and you didnt, I can help and you cant". Especially stepping in and offering to coach, without being asked, in a department that isn't yours... I wouldn't like it if I were the other manager, I wouldnt like to see it if I was senior management.
If your environment allows for and encourages a more open structure, where you're confident you're not overstepping your bounds, yes this would be the tact to take. Considering you have to ask HN if this would be overstepping your bounds, just don't. Just stick to your stuff.
If this isn't an environment where you feel comfortable talking to your coworkers ... then gtfo of there and find a better work place (not directed at you or OP, just in general :) ).
"Business setting" sucks. I'm in a "family-like" work environment where I can approach anyone from CEO to Reception with ideas, feedback, or ask for feedback. It's fantastic. I wish all companies were like this. Unfortunately, that's not the case.
For us, though, he never finishes anything he starts. He does three quarters, then considers it done and ignores requests to wrap it up, so 90% of his work has to be completely redone. That, unfortunately, is baked into his reputation by now, so even if he has perfect execution from now on he would be carrying the reputational baggage he collected over the last years for a while.
If he's been there for a few years and he hasn't changed despite repeated nudges and explicit warnings, yeah maybe he needs to go.
If I'm honest though, smart and always learning isn't compatible with refusing to improve a bad habit. If he's as sharp and eager to learn as you say, it seems like he would be open to feedback.
"Reputation" is quite silly. If the problem goes away and he stays, why the hell should his "reputation" come into it?