Well, I think that it is quite clear from the name and the associated description that the Tree of Life had the purpose of either making the eater immortal or at the least indefinitely extending life (i.e. each eatign suspends aging for a period of time as other mythical trees supposedly did). If the tree had some other primary function and there was no death why would it be called the Tree of Life ? If it were named for a function why not the primary function ? So you are wrong to dismiss it as insignificant.
And it seems quite clear to me that it is BECAUSE Adam ate from the Tree of Knowledge of Good and Evil that God doesn't want Adam to eat from the Tree of Life. Either God didn't consider the possibility that Adam would disobey him, knew what would happen in advance (and therefore knew that Adam wouldn't eat from the Tree of Life before being banished) or the effect of a single eating was only temporary.
hen the LORD God said, "Behold, the man has become like one of (X)Us, knowing good and evil; and now, he might stretch out his hand, and take also from (Y)the tree of life, and eat, and live forever"
Genesis 3:22 NASB
And if you are going to dismiss ideas as beign "unclear" why not dismiss the "no death before Adam" idea on the ground that Genesis doesn't even mention it, the NT references are unclear and the scientific evidence clearly contradicts the idea that there was no death before humans exists.