I would never see it as symbolic in that way - If I was to go down that road I'd do my own head in and be like those women that won't cut the cord at all because they think it shatters the baby's aura in a violent way, so they have to stick the placenta in a bucket of salt until the cord falls off by itself. To my mind, as much of a nature loving hippy as I am - this is far too fussy and inconvenient. I'd rather make my own little 'rituals' and believe that our love and cuddles will help sooth our baby's entry into the world and protect it's aura.
The only reason why it is sometimes nice for the dad to be able to do it is because he can't really do much else to be involved in the actual physical side of birthing, other than give the support etc. The cord cutting is one part of the process that he can do.
BTW Is anyone going to eat the placenta???? (gag! bleh!)