Mine lacks initiative too. He is capable of dealing with pretty much whatever the kids need, he'd just rather do something else... eg rather cook dinner than do bedtime, or rather clean up the kitchen than get them ready for school. So from that perspective he would say he is contributing equally, it's just that he doesn't contribute to the childcare equally, if that makes sense.
His level of general investment in them is definitely a lot lower than mine though. If he has things to do on a weekend day, I'll try to organise a playdate, or a trip out somewhere interesting for them.
If I have something to do, he'll let them stay in their pyjamas and play on iPads most of the day, or maybe take them out for an icecream or to the playground down the road.
I know all their friends' names, and their friends' parents' names, and I organise all their parties and activities. I know when their vaccinations and health visitor appointments are due, I'll notice little rashes that need checking out, or that they might have a blocked ear, or that their hair or nails need trimming. I know when the washing needs to be done each week so there will be enough clean school uniforms. All these things will need to be pointed out to OH.
I don't forget to put their sunblock on, or take their drink bottles when we go out, or brush their teeth in the morning, but he's hit or miss on those things. If the kids call out for help with something, 90 percent of the time I'll be the first one to respond. As a result, these days they automatically come to me for whatever they need, even if I'm in the middle of something and OH is not, which is quite wearying.
The resentment does build up.