My partner isn't helping out enough

No one’s ignored that he’s working. He should still be parenting.

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People are conveniently ignoring that he works 50 hours a week.

But yes on his days off and when he's at home in the evenings he should be co-parenting.

But I wouldn't expect a man who does long shifts to do night feeds etc when I'm on maternity leave.

You both need to change your mindsets that looking after a baby is "work" or "chores" akin to laundry or cleaning the toilets and see it as something you both do when you're at home as part of a team.

And no he doesn't "help out". Helping out is what au pairs do or friends/family. Your DH needs to co-parent.

Does your baby take a bottle?
You need to start doing some hobbies

I agree with most of what you’ve said but this is also problematic, I feel. Dad’s who don’t play their part, when pulled up on their lack of time and effort, will say, “You can go out then”, “ok, fine, I’ll look after the baby while you go swimming/gym”, “I will look after the baby while you’re not here”. That’s not what the OP wants, she wants a partner that is sharing the responsibility of parenting with her.

I see similar advice given often, I didn’t want to go out much when my baby was the same age as OPs (I’m not saying that mothers that do are wrong it was just my preference), and it shouldn’t be that mums are advised to spend time away from their babies as a way to make their partners do more. If OPs baby is breastfed and doesn’t take bottles, I wouldn’t encourage her to change how she feeds her baby as a way to get the Dad to do more. It’s not necessary and further plays into the assumption that Dads can’t do anything to look after a breastfed baby and have to leave it all to Mum, which isn’t true at all.

Part of the problem here is that you and he see him being a parent as “helping you”, as if taking care of his daughter is a favour he can chose to give you or not.

He is a parent. I’m not naive, it can take time in the early days of parenthood to find a balance between parents, especially while one is working and the other is still on maternity leave. When my husband is home we parent 50/50 and I believe that’s how it should be. If I’m cooking dinner, he’s playing with the kids, if I’m reading bedtime stories he’s tidying up. When my babies were small they were easier to comfort in my arms and often wanted me when they were upset but this only gets better with time and effort from your husband. He can bathe her, wear her in a sling, take her for a walk in her pram.

We don’t count minutes but we have fairly even free time each. When my babies were small and I didn’t really want to go out without him this might mean an extra long soak in the bath or an hour upstairs reading/watching Netflix alone.

Remember, while he’s working, you are too, looking after your baby. He needs to remove the idea from his head that because he works you take responsibility for the baby all the time. You need to spell this out for him now since he hasn’t got that worked out for himself. Be firm and tell him you refuse to parent with a partner who doesn’t see you both as equal parents!

1. it's not "helping out". That's the first problem. He's thinking about how much he feels he's given up. Not looking at what there is to do and 24 hours in the day and how to make sure the load is shared fairly.
2. Anyway, I'm worried our relationship won't survive if he doesn't start have you spelled this out to him?

Every hour he spends gaming is an hour that he is relying on you to look after the child you had together

Does your baby take a bottle?
You need to start doing some hobbies.

Is this relationship with a gaming man child worth saving?

Tell him this.