The judge needs to consider everything, but the history is also very important. Does it matter if an NCP wants to try? Possibly, but of course, the issue of visitation wouldn't really be on the board if he didn't at least claim to want to try. But, a "try" sounds very wimpy (excuse the jargon) - a judge could say, "listen buddy, these are your children, a mere try, which implies you are prepared to fail in your efforts, is not in their best interests. They need a parent who WILL be there, without fail, who puts them first, cherishes any time he can get to spend with them, and understands exactly how important they are..." etc. etc. So, if I had a deadbeat who had blown off his children for years, and now out of the blue wants to "try", I would likely ask the judge to deny him, due to the fact that he had previously devastated his children by never being around when he could have been, causing them repeated heartache and pain - and only to allow supervised visitation
in originally limited quantities, after he had shown he had stabilized, was not doing drugs (if every he had that problem), had a stable home and a stable job for 1-2 years. Perhaps phone calls and skype at first, then if he shows appropriate lifestyle for 1-2 years, some slowly growing visitation.
In a perfect world, he'd change his life around, gain trust in his kids, and there'd be a terrific father/child/children relationship forever more - for the good of the kids, who we all know do best when they have two loving and stable parents in their lives. But, if he doesn't or can't, I think his involvement can harm more than hurt, particularly if he constantly lets them down, lies or breaks promises to them, etc. It may be that it can damage their self esteem, teach them to not trust (which can lead to future unhappiness and failed relationships later in life), etc., and may not be in their best interest. For this reason, him having to prove himself as having straightened out, may be something a judge would demand he do before giving him visitation.
Good luck, they are lucky to have you!