Interesting view, although I'm not a Christian myself but I like studying different religions and have read the bible few times. For me this part of the bible always seemed little too fairy taleish compared to other religious view on the beginning of mankind.
We have to ask ourselves is death a bad thing? What would happen if there was no death on earth? Also if we assume that Eden is a place that exists somewhere other than earth what would happen if there was no death in Eden as well?
And if we assume that Adam didn't get punished and sent to earth but instead stayed on Eden and started reproducing and humans started growing on Eden then unless the Eden has infinite space it would eventually run out of space like earth.
So from this point it is clear that death is a necessary process if there is to be any kind of life to exist anywhere in the universe unless Star trek turns into reality.
However all this assumption is purely from a pragmatic and materialistic view and if we consider the fact that every living thing can exist not only in material form but also in spiritual form, another words if what we call death is actually not The End but just another form of living then it can be interpreted in an entirely different way.
But then the real Truth can be very simple :)