The story does not say that the Tree of Life was for eternal life. When you inject 'eternal', you are going beyond what is written. However, I think religious leaders imply and count on you assuming certain things that just are not in the story. And that holds true for many other stories as well.
The Tree of Life is the opportunity of living on earth as a blind sheep, happy to just graze in the garden, not minding when the Masters take your wool and leave you cold in the winter. Some of the sheep did not appreciate being fleeced all the time. They wanted to know why they were being used and abused. Eve was such a sheep.
Adam and Eve worked hard in the garden, only to have some man claiming to be God, take all of the fruits of their labor. Eve was not content to break her back all day in field service, even though she was permitted to enjoy TV at the end of the day and hot tub on Saturday's only. She wanted to know WHY. What was it all for? Why did God not have to work? Why did God get the hot tub everyday? Why did God get all the money?
That's why she chose to eat (eat=learn; discover) from the Tree of Knowledge. She knew there was much more going on than what they were being told by 'God'. She was not content to be a blind sheep slave captive anymore. Tree of Knowledge was a quest for Truth.