Your DS is 4.5 months 8DaysAWeek?
That's still tiny, only just out of the newborn "fouth trimester" stage. If you are breastfeeding then evening cluster feeding (so just spending your evenings sitting on the sofa with baby in lap napping and feeding on and off) if normal. If you are formula feeding, feeds may still be frequent through the evening and baby may well need to comfort suck a dummy to help with the evening unsettledness.
he's entirely dependent on ME getting him to sleep
He is entirely dependant on you to get to sleep. He may well be for many months to come and that is a massive responsibility.
Yes, Dads can take a role. But assuming you are primary carer (ie the one who's Home with baby all the time) then in these early months it will be you who baby wants for comfort to sleep. That's likely to be the same if you didn't breastfeed, but obviously you could share the feeding responibilities more.
But a tiny baby (under 6 months, probably closer to 12 months) will indeed be imprinted on whoever their primary carer is and since usually it is Mothers who have the parental leave, that is usually mum.
That's a huge responsibility and I'm trying my best
No one here doubts you. It is hard.
Speaking personally the single thing I found hardest with my first child (I have four now) was the dawning realisation of the level of responsibility I had to bear. And the realisation that I had very little power or control, having to just accept the responsibilities for baby in whatever form the baby presented them to me, and cope. I suppose the "giving myself entirely over to being a mum" was what I struggled with.