Only so long as it didn't get too cold for heavier elements to form could helium survive.
Big Bang nucleosynthesis produced incredibly few stable nuclei of elements heavier than lithium because of a bottleneck brought on by the absence of a stable nucleus with 8 or 5 nucleons. Additionally, this lack of larger atoms limited the quantity of lithium-7 that could be produced during BBN.
The early cosmos was too hot before a few minutes into the expansion for atoms to form. Only a few minutes remained before it became too cold to continue nucleosynthesis and prevent the formation of new elements.
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