I don't think he gets to use his depression as an excuse when he's not doing anything to actually improve his situation.
I get that it can be hard for people (especially men) to get help when they're struggling. But I couldn't overlook his lack of action to repair his relationship with his child.
Is he the sort of dad who thinks throwing money at someone can fix a relationship? Was that why he stole from you? Because not only is that shit, lazy parenting - it's also theft. From you. The person who he supposedly wants to marry and share finances with forever.
To put it in terms of employment. Most employers have a disciplinary process which is different depending on what an employee does wrong. Minor infractions such as repeated lateness, making repeated mistakes in work, being a bit rude to customers/colleagues - those are misconduct and would probably draw a recorded verbal or written warning. Get three of those and the employee is out the door. Or, a major issue - defrauding the company, calling a customer a cunt, smacking a colleague in the mouth, turning up drunk - those would be gross misconduct and grounds for immediate termination.
If an employee does something classed as gross misconduct, their previous history is irrelevant. If they are on no warnings or on their final warning, if they do something severe, they will be terminated without notice.
What you're doing with the "3 strikes" thing is treating his previous conduct "issues" as minor mishaps, for which you've essentially given him two (3?) warnings. And if he makes another, then he's out the door.
However, the shitty parenting and the theft, for me, would be gross misconduct and I'd have binned him off immediately for the latter.
You may view his shitty parenting differently. But if you are wanting children with a life partner, to me that alone would be a deal-breaker. Because why would you want children with someone who's prepared to give up on their child at the first sign of difficulty?