TL;DR Check if they have full neck mobility and didn't get too strained in the birth (if natural), and if they have a tongue tie sooner rather than later.
https://www.nct.org.uk/information/baby-toddler/feeding-your-baby-or-toddler/tongue-tie-babies
Loads of other advice here and falling sleep on the breast sounds like a comfort thing but I'll share some recent experience on feeding issues just in case.
Baby had a mild tongue tie, fed on the breast the first few feeds weirdly enough but after then they would get frustrated as they couldn't latch properly.
We had a lot of luck with silicone nipple shields as instructed by the midwife, which helped with latching but no luck retiring them, so they became permanent for a while (they're recommended as temporary).
Even after that we still experienced a lot of fussing and stress with feeding which put a lot of pressure on my SO for the same reasons you mention.
Doctor recommended we go to an osteopath to get their neck and back massaged because getting squeezed out during birth had put a lot of pressure on the neck and restricted turning the head which also caused feeding issues. After this their tongue was a lot more mobile and feeding got a lot easier because they use their tongue to support latching.
Unfortunately this was already 2 months in and they were pretty hooked on nipple shields/bottle feeding at this point so they were still stressed about natural breast feeding which felt like it was a lost cause to go back.
Finally after lots of back and forth with different opinions from medical professionals, lactation consultants etc the agreement was that the baby had a mild tongue tie and we decided to get the frenulum cut/lasered which gave more tongue mobility but at this point the baby seems to have forgotten that milk comes from the breast so after almost 5 months we have just stuck with bottle feeding and pumping.
I'd suggest ruling these out ASAP as it was pretty stressful and especially left my SO with feelings of inadequacy not being able to breastfeed. Hope it helps.
A curious and questionable, watery space.