IIRC, the essence of keeping the first couple from the Tree of Life was so that they could not ‘continue’ to eat from that tree.
You may be correct. There is nothing said about the tree of life before Adam and Eve were banned from the garden. However what is meant by “life” here? If it means eternal life, you need eat it only once. You need to realize that eternal life cannot be conditional. Anyone who dies, did not ever have eternal life.