There's a good deal of additional information that would help but, from what you have posted, I'd say your prospects are slim to none.
Take a look at the case file and see how you were served. The mailing could have been only part of service. In any event, at some point, the court had to find that there was proper service or it would not have entered the judgment.
The same reasoning should apply to the debt. The plaintiff must support its case to the satisfaction of the court, whether you oppose it or not. It's called the Burden of Proof, and it's the plaintiff's responsibility. That suggests to me that, at some point, the court was satisfied that there was a legitimate debt, that it was not beyond the SOL, and that you were properly served and had a chance to defend but didn't.
Finally, despite the fact that nothing has been collected, what has happened since judgment was entered? Was the judgment recorded? Was an Abstract filed? When did you learn about the whole thing? Courts have a bias in favor of a "hearing on the merits", as opposed to a default. So, for example, if you had been properly served under Ohio law but it just happened that you were traveling outside the country at the time and the case went to default before you actually found out about it, your court is likely to be reasonably generous about vacating the default and giving you an opportunity to defend yourself. BUT you must act promptly, as soon as you learn of the judgment. You've waited two years. I don't know, but I'm inclined to think that you didn't find out last week, and you've waited too long to even file the motion.
So, unless you have all the facts falling the right way for you, who's to say that the attorneys have to show up? They could file a written opposition pointing out all or any one of the foregoing problems, and the court could deny your motion on its own. I suspect that it would.