First point - the tree did not give access to eternal life. The tree meant man could live forever. Eternal ‘life’ does not equate to ‘living forever. This is crucial, as I will explain soon. But first, an understanding, one that may not necessarily be accepted by all - so it is for consideration...
All live forever. Scripture clearly alludes to this. Although there are some who believe in Annihilation - Nevertheless ...
MAT25:41 Then he will say to those on his left, ‘Depart from me, you who are cursed, into the eternal fire prepared for the devil and his angels.
Those ‘on his left’ are people! They are going to the same place as the devil and his angels
REV 20:10 And the devil, who deceived them, was thrown into the lake of burning sulfur, where the beast and the false prophet had been thrown. They will be tormented day and night for ever and ever.
Note - for ever! Everyone will ‘live’ eternally, but there is a difference between ‘life’ and ‘death’. Death does not mean not ‘being’. So what does it mean.
Here we run into a conflict between the dictionaries definition of ‘death’, and the biblical definition of ‘death’. Biblically, death means separation. In our (western) understanding, we [only] consider ‘physical death’. But ‘biblically’ physical death is separation of your [physical] body from ‘you’ [spirit/soul].
‘You’ will still be ‘alive’ after your [physical] body isn’t. [breathing]. But this ‘view’ is not ‘universal’ - so let’s just put that aside and look at your question from another angle...
Why didn’t God want Adam to be able to ‘eat’ from that tree?
GEN 3:22 Then the Lord God said, “Humans have become like one of us; they know good and evil. We must keep them from eating some of the fruit from the tree of life, or they will live forever.”
The ‘Bible’ reason is so that man wouldn’t live forever. You ‘interpret’ this as punishment. That not what is says. Nevertheless many argue that it is clearly ‘inferred’. But let’s consider scripture...
God said ...
GEN 2:17 [snip] for in the day that you eat of it you shall surely die.
Did Adam die? Depends on whose definition you use. The dictionary says no. But, the Bible says yes! Now - remember ‘death’ [biblically] means separation. The day Adam ate - he [spirit/soul] were instantly separated from God. His ‘body’ didn’t ‘die’- his ‘body’ could still communicate. But ‘he’ was spiritually ‘dead’ [separated]. Separated from the One who gave him ‘Life’. So ‘he’ (although ‘alive’] - was in fact ‘dead’ - just as God said would happen if he ‘ate’.
1 COR 15: 22 For as in Adam all die, even so in Christ all shall be made alive.
Now let’s put this together - Adam was dead (spiritually). Because ‘death’ Biblically means separation, and Adam’s spirit was now separated from God. And if he could now eat from that other tree - he could ‘live’ forever. After all, it was the ‘tree of life’. But he would ‘live’ forever spiritually separated. He (man) could then no longer ever be reunited with God. Man would be ‘forever’ separated from God. Tragedy! This had to be prevented - and must not be allowed to happen. So God removed Adam from being able to access that tree - until ‘man’ could first be reunited back with God -
The expulsion was an act of Love - not punishment. The ‘tree’ of Life refers to physical life. The consequence of Adam being removed was physical death. Man must not be able to ‘physically’ ‘live’ forever separated from God. That would result in ‘hell’ on ‘earth’. A Loving God would never allow that.