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.
I believe the phrase from Genesis refers to ‘eat thereof and live forever in their fallen state.’ A man without God’s life in their human spirit is dead. A man with God’s earnest of the inheritance in his human spirit is alive forever more because God Life is unending. The ‘live forever’ relates to the ‘fallen state’, thus we know God made humankind to have unending existence, but without God Life in the human spirit, that living is comparable to being dead in trespasses and sins. Adam and Eve would be around in one form or another even beyond their physical bodies. Eating from the Tree of Life would have extended their physical lives unending, but in a fallen or godless state.