The Hebrew language is often under-determined (i.e., there may be several viable options for interpretation)–especially in an under-determined text like Gen 2-3–and, as noted by others, translations are limited (e.g., philosophy of translation, variance in semantic and syntactic ranges between original and target languages, interpretation of the translator). Relevant here is the difference between Hebrew and English concerning verbs. English is heavy on verb tense (past, present, future), while aspect (i.e., how the action is being viewed, especially in relation to other actions) is usually noted by additional endings (e.g., -ing for gerunds) or additional words (e.g., has been running, had eaten). In Hebrew, however, while tense is not absent, aspect is more dominant, and tense often has to be inferred. Thus, an imperfective verb–like the chain of imperfective verbs at the end of Gen 3:22 (it’s actually more complicated than this, but I’ll try to keep it simple)–could be understood in various ways (e.g., eat ever, eat from now on, eat continually or intermittently, etc.).
I think we need to separate two issues (answers to which must bring in multiple factors): (1) Did A+E eat from the tree before this moment?; and (2) What is being prevented in the future by removing access to the tree?
On #1, I don’t think it’s clear (nor does it matter!). On one hand, the tree of life was one of the multiple options, so one might assume they had eaten from it (but, then again, we don’t know how long they were in the garden). On the other hand, nothing says they had (and Gen 3:22 could certainly be taken to mean they hadn’t). But does it matter? I suppose it would matter if eating once from the tree resulted in irreversible immortality. But nothing in the text demands this (certainly not the verbs in Gen 3:22). [BTW, the rabbis had a field day wrestling with all of this!]
On #2, it seems clear the exile from the garden was to keep sinful A+E from living in a perpetual state of sinfulness. But what would’ve granted them this sustained state (i.e., immortality)? One could point to the tree itself (having some sort of magic powers)…then one might ask why God just didn’t remove the tree. Or, in concert with the rest of Scripture (IMO), “life” is granted to those in the presence of God. This would allow the tree of life as a means to stay in God’s presence. Having been created mortal, loss of access to the means to stay in God’s presence resulted in A+E to suffer their natural mortal state. (Thus, the idea of continually needing to eat of the tree would’ve been necessary.)
Of course, there are larger issues at play, such as genre. That is, while the narrative is about trees and a talking, sinning snake and a strolling God (and other “gods” with the “us”), we still need to ask the larger interpretive questions about authorical intent with these references and symbols. But that’s beyond the discussion here.