A few things.
You say "He constantly apologizes for taking my time" - this may be because he's picked up on your own feelings and irritation about the matter, and can see that you view these interactions as an undue burden (at least overall, if not in any one case assessed individually).
You say "[he] checks in a lot to ask me if what he's about to do that day will be useful." - on the face of it, and without more detail about exactly what he is asking for or doing, this may simply be an attempt to facilitate your role. If your role is technical and specialised in nature, he may see himself primarily as a facilitator rather than a dictator, and is looking to you to provide some indication of what facilitation is required.
You say he doesn't feel that he carries his weight nor understands your role. Does that actually ring true, or does he seem adequately competent? Is there a radical age difference between yourselves?
One possible overall explanation is that he is becoming ill with depression - perhaps because he can see that he is a fifth wheel in the activity at work, and isn't content to fill a sinecure position, but also perhaps because of circumstances in his personal life.
Another possible explanation is that you are in fact the natural leader and most competent operator in the room, and your manager is deferring to that. This is perfectly possible if you are an outstanding performer, or alternatively if he is approaching the end of a career where, for whatever reason, he has not maintained the currency of his technical skills (or has arrived in management via different roles than the ones he is now expected to manage).
I'd be inclined to closely examine claims like "it creates a toxic team culture", or that his behaviour "will lead to poor delivery". Is there actually a toxic culture emerging involving others in some way (the details of which you don't articulate), and is delivery actually becoming poor? Or is the main problem so far merely that you alone find his behaviour toxic?
Is the strain on you real, in terms of placing a significant burden on your working time, or of creating informal responsibilities with a large cognitive burden, that can't coexist with carrying out your formal role?
Or is the strain merely in terms of finding his behaviour unholy and un-manager-like in some way? Some people prefer to participate in strongly hierarchical relations, rather than more collegiate relations.
I'd think about whether you can find something useful for him to do, some way to involve him more in the work, or whether you can otherwise set the terms of the relationship in a way that you determine but which is mutually satisfactory. Effectively, adopt part of the managerial role which you think he is leaving somewhat vacant, and cooperate in the management. If you intend to go further in the firm, you may then have a strong and trusted advocate for your progression.
If you suspect he is becoming ill, then maybe solicit some more information about whether everything is alright? And perhaps approach another benevolent manager in confidence for advice about the situation.
In the final instance, you could ask for a transfer to another team on a pretense, or seek another job externally. But I wouldn't stick the knife in and make a complaint about his behaviour and competence, or talk in terms of "toxic culture", unless he is causing chaos in the team of a kind that is pre-eminent and can be seen externally.