Well, every birth is different. I never felt left out at all. My IFs actually wanted me to nurse in the hospital so I did. Once baby was born I held her first and she nursed for a bit. Then I basically got to introduce her to her daddies. The next hour or so was all about them. Pictures, snuggling, etc. I was getting stitched up and whatnot but it was awesome seeing them as a little family.
IFs and I had our own hospital rooms but they were side by side. Since I was nursing they would text every 1.5hr-2hr to come in to feed the baby. This was a bit overwhelming since I was naturally exhausted. Normally the IFs would hang out while baby would eat and we'd talk. IF's mom was also there so it was nice getting to know her. They would read all the comments that their family had sent about the baby, which was nice seeing how loved she already was. I was alone with the baby a few times. I remember at one point they'd all gone back to breakfast and the nurse came by my room because baby was hungry. I immediately texted IF just to let them know where she was and they came up a bit later.
I was never really itching or needing to hold the baby. It was honestly a great feeling to see them all together and how happy they were. They were discharged about an hour before me and returned home. Lots of hugs and goodbyes. We still have a good relationship and exchange regular texts and pictures. I haven't seen baby since the hospital but I have an open invitation to do so (they live about 2hrs north). Just haven't really felt the need to do so. I know she's in good hands.
So, no I never really felt discarded. I do think there is a major attention shift and (for lack of a better description) change in control after baby is born. You go from having all eyes on you...texts, calls, emails about when baby is coming, etc to all eyes on baby. I think there's a shift with EVERY pregnancy but we don't notice it as much since we have a baby to care for. So it definitely feels a bit more pronounced with surrogacy.
I don't think it's weird. Just keep in mind everyone is different. What may be normal to one may not be to another.