I never wanted to hand my babies over (especially to other people's small children - why did people think I should give my newborn to their 3 or 4 year old to hold? This hit its peak with DC3 whom people seemed to think I should casually chuck across the room at any passing toddler and be bubbling over with gladness to be rid for as long as possible...).
I've never got why everyone expects newborns to be handed around like parcels, and that goes in triplicate when the visitor doesn't want to hold the baby but feels they ought to!
I also don't especially want to "have a hold" or "have a cuddle" with other people's newborns, though I am more than happy to hold any baby if the mum does want her hands free to drink her coffee or go to the loo!
I do think its polite to show an interest in the baby when visiting, as clearly the arrival of a new baby is the "big event" in the household, whether it is interesting to you or not. In the same way if somebody has just got/ is just about to get married its polite to show some passing interest in the wedding talk/ its polite to ask how somebody's holiday was if they've just got back from a special holiday (especially if they only go on holiday rarely rather than multiple times a year) etc. etc.
Expecting a gift is grabby but on the other hand it is something of a convention if - and only if - the visit is specifically to meet the baby (not if thy are visiting to drop off Christmas presents - expecting a baby present too is really grabby in that context)
I like babies but neither as mother of newborns nor as visitor to parents with new babies do I get the compulsion to hand them about!