From what I read your wife suffers from depression, correct? From what I read between the lines, she already did before she got pregnant, correct?
And considering how experienced you seem with taking over and keeping everything together, that’s probably what you have been doing for years already?
That’s quite a common pattern, and it’s one that can only remain stable for a certain amount of time. She’s depending a lot on you, you pick up the slack and carry her burden. That works well without kids when the only responsibilities are to spend enough time at work, but it becomes very troublesome with a small kid, where the workload is too much even for two fully-functioning adults.
This can drive you in a kind of caregiver burnout. You go beyond your limits for too long, and after some time you just don’t have the power to continue that way and smile through it. Depression spreads and good things diminish. That’s at least what I read between your lines.
This is the point where you need to get help. Find a better distribution of work with your wife. Rope in relatives (your mom seems to be invested) and get them to help you out, especially in these crucial first few months.
Pumping milk means that the baby isn’t necessarily tethered to your wife, so you can also get your mom to watch the kid for an evening or so, so that you two can get some rest.
Considering getting therapy yourself.
Try to recover before you burn out completely.
As for the feelings towards your child, don’t force it, give it time.
It’s not so much of a dad-vs-mom thing, tbh.
For me it was super easy to connect to our kids especially when they were tiny. I loved carrying them around all the time, the cuddling, them being as cute as they are. My wife really hated being touched all the time and she couldn’t connect with them at all in the beginning, especially with our second one.
I started having more and more trouble with the kids when they got old enough to have a mind of their own, especially with our first kid who was and is much more than just a handful. My wife gets along much better with the kids once they are able to talk and able to take more care of themselves.
Some people just connect with the small ones better, some can handle them better when they get bigger. Some then start having issues with them in puberty, while some really manage to connect at that time. It’s not a gender thing at all.