Unsupportive partner/father

Might be a bit drastic but maybe take a day for yourself and have husband look after him and he will see how much work it takes to care for your little boy and maybe he’ll be supportive afterwards xx

From what you've said he doesn't sound like he's a bad dad, but rather that he's in a bit of a rut.
Like Emma said give him no choice and see if he rises to the occasion. The fact you addressed it with him and he reassured you he'd try seems like a positive step. Unfortunately at first it might be down to you to guide him on what you want him to do. Better to do it now then so he can try and get back to good habits.

I also think as sexiest as it sounds us women naturally know how and what to do when it comes to the children even in times we feel lost and don’t know what to do. It took my husband a while to know how to be helpful with the kids other than being fun dad but then I’d ask him to step up in other ways where he felt more confident, I would ask for help with the house the dog and other tasks Instead. I also think playing to each other’s strengths instead of trying to be equal also helps. And like others say maybe start off with an hour or 2 by himself and work with him to build up that confidence. Half the time they are scared to get it wrong, I know my husband was as he knew I had all these routines rules etc he didn’t want to mess up. You are a team, and it’s a great step he is being reassuring however men do tend to need a bit more direction. Xx

I would divvy up what feels fair and present it to him.
Husband I'm so glad you're said you'll step and support our son because I feel unsupported.
Alternating nights of showering and put downs, xyz, xyz, and x,y,z. I feel this is fair shall we start from Monday? So glad you're onboard.

Then it's his job to do the shower on whatever days you've agreed and you don't ask if he owns it! (X,y,z add other bits for him to own)

You shouldn't have to do this i.e. the mental load but this is what I'd do. Be explicit then you hold him accountable to actions not just words which are just passifying. Xxx

For us. It’s come to a point that I’ve written a schedule and it’s just non negotiable. No need to discuss anything everyone knows their time and what they’re doing