He's pulling the wool over your eyes OP. He's opting out of childcare because it's boring.
Yes, this.
People are being a bit snappy/harsh - please try to see past this and not take it personally! I think it's because this pattern is all too common, and often excused as it being harder for dads or whatever. It's not harder. It's hard (bloody hard) for mums as well. It's just people sexistly assume it will be harder for dads so they get a free pass, which isn't quite fair. It's frustrating to see the same patterns being played out.
A lot of parenting/childcare is boring. This can be a bit of a shock to the system. You said that your DH comes from a large family and so having children was a no-brainer to him. That was me a bit as well. I didn't come from a large family but I was always envious of large families and loved the idea of having lots of kids running around the place being chaotic and funny and messy and loud. I think when you're assuming it will all (or mostly) be fun and that the hard parts are completely manageable and when you focus on this fun tumbling mass of children you are actually picturing the fun that it is being a part of the children group. You forget that in fact it's very different being the grown up! I don't think anybody is ever prepared for having children but I think it's a special kind of shock when it's something you've just always known you would do without really considering not having them. And everyone always talks about the positives but not the negatives which can also skew things and make you feel like you're getting it wrong if you don't enjoy every second. It can also feel really overwhelming to suddenly be the grown up and be in charge.
The thing is - it doesn't really matter if he thinks he'll be better at the bits when LO is a bit older. Perhaps that's true, it might well be, but if he doesn't put the foundations down for the relationship now, then he'll struggle to relate to LO when they can do those things as well. You can't just skip the bits you don't like.
I think you should talk to him and basically say that he needs to figure things out. Yes he will make mistakes and get things wrong but that is part of parenting, it's part of the learning curve. It doesn't really matter if he doesn't get things perfect because there is no one right way to parent anyway. After seven years of this what I've worked out is that there are only really three massive fails in parenting: Verbal/physical abuse, having literally no boundaries (ie giving into EVERY demand ever), and neglect of their needs (enough food, water, basic awareness of physical safety, etc). Everything else is up to you as a parent. And yes everyone will have an opinion and you'll always be getting it wrong in someone's book. That's just the way it is.
You need to be able to rely on him as a co-parent and not have everything fall to you. It might be manageable now, one year into parenting a single child, but you don't want the situation to stay the same for the next 17 years, or once you introduce a second child into the mix. It's just not sustainable, and it's not fair. What if you have to travel for work? Or just want to go on a hen night or something? What if you were ill and had to go into hospital? Or care for a sick relative? You need to know that he is capable of stepping up in these situations, because you don't want to be stressed out further by worrying about your DD.
He will get better at caring for her alone if he cares for her alone more often. Push him out of his comfort zone a bit but let him decide what to do. He doesn't have to go out and meet people if he doesn't want to - taking her to the park is just fine. I'm sure he can google for stuff to do just as well as you can, so let him find his own way. I get that you're trying to be supportive, but I think that by offering all this help you're shooting yourself in the foot a little bit because he's always expecting you to when really he needs to be thinking about things himself and working out what to do, and it will only get harder the longer you leave it. He will survive if he goes swimming with her alone.
It's probably better to word it in terms of "I need" rather than "You should..." "Other dads..." etc because that can come across as controlling or like you're saying what he's doing now isn't good enough which isn't what you're trying to say.
I'd word it something like this: "I need you to start taking LO out alone sometimes so I can get a break." "It would be really helpful to me if you could plan activities with LO without my help". "I might not always be here to help you out, so it would reassure me if you can get more confident with spending more time with LO alone". If he asks for suggestions then fine, but try to sit on the temptation to offer them unless asked! And even then I'd try to lean towards suggestions that promote independence like suggesting he ask his parents or siblings for advice (if siblings have children) - after all his parents raised lots of kids! - or suggesting he googles things. If he likes web forums reddit has a forum for parenting and one for dads specifically. But really, the crux of the issue is you need to back off and let him take responsibility for their relationship, even if that backing off is a bit more like shoving him out the door than it is you tiptoeing away.