Why is Big Bang Nucleosynthesis favored over Stellar Nucleosynthesis? We know that a hot soup of plasma as it cools (as space expands) can produce light elements if the baryon-photon ratio is just right and the space expands quickly enough so that the neutrons and protons fall out of equilibrium - a theory advocated by Ralph Alpher.  We also know that Deuterium and Helium can be produced by fusion of hydrogen atoms in stars, a theory advocated by Fred Hoyle.  The internet is full of articles that tell us that BBN is one of the pillars of Cosmology, but I can't find a single one that says why it's favored over Stellar Nucleosynthesis.
Please explain the reason/evidence for preferring BBN over Stellar nucleosynthesis for the production of light elements.
 A: Deuterium and lithium are generally destroyed by nuclear reactions inside stars. They fuse at lower temperatures (about $10^6$ K and $3\times 10^6$ K for D and 7Li respectively) than the $>10^7$ K it takes to convert hydrogen into helium.
Therefore, if the conditions are right to produce helium from hydrogen, then this should destroy all the deuterium and lithium.
NB. 7Li must be created somewhere because the cosmic abundance is higher than the primordial abundance. The most likely candidates are AGB stars or novae. But then 7Li would be accompanied by lots of other heavy elements and wouldn't explain why we see lithium in extremely old stars, even when they have almost no metals.
It is also worth mentioning that the exquisite concordance of the measured primordial He and D measurements at a single value of baryon/photon ratio that itself matches the value estimated from the cosmic microwave background (for example see this "Schramm plot") is a coincidence beyond any variant of the steady state theory.
