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If heavy elements were created in the cores of first-generation stars, and this star's been around for the entire time, wouldn't it have the same concentration of heavy elements as the rest of the universe?

Good question; according to the models for stellar energy production, a star such as the one in this article evolves very slowly compared to larger stars, and the production of haevier elements always occurs at the end of the star's life cycle, after its hydrogen fuel is used up. This star is still in its hydrogen burning phase, and has not had an opportunity to produce heavier nuclei than helium.

Additionally, at 0.8 solar masses, it might not be massive enough to EVER get hot enough to initiate helium fusion, and hence will never produce anything heavier than helium.

Lastly, truly heavy nuclei (>Fe) are never produced in normal stellar reactions; they are produced in supernovae.

The weak force is the force that induces beta decay via interaction with neutrinos. Not only would the Sun not burn without this force A star can “burn” by a nuclear fusion process. Three of those processes are proton-to proton fusion, helium fusion, and the carbon cycle. Here is an example of proton-to-proton fusion, which is the process our own sun uses: (two protons fuse -> via neutrino interaction one of the protons transmutes to a neutron to form deuterium -> combines with another proton to form a helium nuclei -> two helium nuclei fuse releasing alpha particles and two protons). The weak force is also necessary for the formation of the elements above iron. Due to the curve of binding energy (iron has the most tightly bound nucleus), nuclear forces within a star cannot form any element above iron in the periodic table. The curve of binding energy comes from the strong and electromagnetic forces. The role played by the weak interaction is to convert protons to neutrons and vice-versa, which is often required to make stable nuclei out of two lighter ones. So it is believed that all higher elements were formed in the vast energies of supernovae. In this explosion large fluxes of energetic neutrons are produced which produce the heavier elements by nuclei bombardment. This process could not take place without neutrino involvement and the weak force.