I agree that people are over reacting to the behaviour of OPs partner. I dont think he's up for Dad of the Year, but I dont think I was up for Mum of the Year when DC1 was 5 weeks old either. I was tired, frustrated and a pretty lonely too.
Op,I would suggest that if DP doesnt want to deal with the crying that you have to make your decision. Do you want to push the issue, and take a step back so he has to deal with it. Or, do you want to calm the baby while DP does his bit to support you by doing the cooking, washing, cleaning, making you lots of cups of tea etc. It is hard to calm a screaming baby, and failing at it constantly while you succeed is probably demoralising for your DP.
I breastfed so when ex was here he would say there was nothing he could do to help. So, eventually I made a list of all the helpful things that he could do for our family other than feed the baby.
Like Ginger, ex-DH always did bathtime, and also did winding and settling after a feed whenever he was at home. At weekends he took DS for an walk in the mornings for about an hour so I could have a lay in, and then another in the afternoon so I could get my feet up for an hour. Ex wasnt around much for the baby stage with my youngest DC and I realised how much he did.
If he enjoys the baby book and taking photos then try to build on this. For example, has your son been to the local woods or farm? Could you and DP take him and DP be photographer for the day, so he can capture these family moments for your DS book. My DC always loved going to the woods and spent ages looking at the light coming through the trees.
Dont give up on hiim, but also dont give up on trying to improve the situation. It will get better.