Without taking on any of his debt or cleaning up any wreckage from his addiction, in my opinion I would step in and do the little things with his immediate financial situation to help it get stalled like you thought of.
Making phone calls or helping him figure out his insurance to get them to pay for rehab. may be beyond him at this point in time,. Luckily, what he is capable of is getting help.
Once he is stable he can then take back his responsibilities.
He is willing to go inpatient and work on his addiction, it is okay to let that be his sole focus for awhile so he has a chance at recovery. If he is the father of your child
you hope he gets well whether you go back to him or not.
Do what feels right.
Go to al-anon for yourself.