Why are lithium-$7$ and boron-$11$ respectively more abundant than lithium-$6$ and boron-$10$? Lithium, beryllium and boron are unusually low in abundance for low-$Z$ elements, because their stable isotopes aren't the ones stellar nucleosynthesis makes without consuming them. What little abundance they have is primarily due to spallation. The stable isotopes are ${}^6_3\text{Li},\,{}^7_3\text{Li},\,{}^9_4\text{Be},\,{}^{10}_5\text{B},\,{}^{11}_5\text{B}$. I couldn't find the reactions that form them, so I've tried making educated guesses viz.$$\begin{align}{}^2_1\text{H}^++{}^4_2\text{He}^{2+}&\to{}^6_3\text{Li}^{3+},\\{}^1_0\text{n}+{}^6_3\text{Li}^{3+}&\to{}^7_3\text{Li}^{3+},\\{}^2_1\text{H}^++{}^7_3\text{Li}^{3+}&\to{}^9_4\text{Be}^{4+},\\{}^1_1\text{H}^++{}^9_4\text{Be}^{4+}&\to{}^{10}_5\text{B}^{5+},\\{}^1_0\text{n}+{}^{10}_5\text{B}^{5+}&\to{}^{11}_5\text{B}^{5+}\end{align}$$(add photons to taste). In particular, the above choices avoid tritium due to its rarity.
Lithium is $92.5\%$ lithium-$7$; boron is $80.1\%$ boron-$11$. On my hypotheses, in both cases the more abundant isotope is made by adding a neutron to the less abundant one, implying most of the lighter nuclide fuses with neutrons. Are there enough neutrons (today or in the past) to explain this, or have I got the reactions wrong? I'm sceptical because you'd only expect most of a nuclide to fuse with neutrons if the latter's concentration is very high, but this article claims the spallation responsible is post-primordial.
 A: Lithium
Actually, only about 20% of present-day lithium is cosmological in origin. This is made by adding free neutrons to beryllium-7. The rest is mostly made inside stars and probably in novae (not supernovae) explosions (e.g. see this popular account), not in spallation reactions.
The article you cite is almost certainly incorrect. There is plenty of direct observational evidence for Li-rich giants and for novae with signatures of beryllium-7 (which decays into lithium-7) (e.g. Monaco et al. 2011; Tajitsu et al. 2015; Selvelli et al. 2018; Singh et al. 2021).
The basic reaction there is to add an alpha particle to helium-3 to make beryllium-7, which then decays by electron capture to give lithium-7.
The trick is to make sure the beryllium or decayed lithium is removed from the high-temperature site of its production before it fuses with protons to form 2 alpha particles. This latter reaction is why lithium is rare. Protons are usually abundant and this strong-force reaction occurs readily at temperatures below that of hydrogen fusion. In novae, the beryllium-enriched material is carried away in an explosion. In giant stars it may be something like the "Cameron-Fowler" convective mixing mechanism (Cameron & Fowler 1971).
Lithium-6 should hardly be produced at all by primordial necleosynthesis. The main reaction is a "radiative capture" reaction of a deuteron and an alpha particle. This itself has a comparatively low rate because of its electromagnetic nature, but also lithium-6 is readily converted to alpha particles by reactions with deuterons.
Thus any lithium-6 in the universe is post-primordial. But lithium-6 is even more fragile than lithium-7 inside stars or other nucleosynthesis sites because it has a lower binding energy per nucleon. It can be destroyed by fusion with protons to make lithium-7 and this occurs at slightly lower temperatures than the fusion of protons with lithium-7. Thus post-primordial production of lithium in stars favours the heavier isotope.
Spallation may be a smaller contributor to post-primordial lithium production (but much more important for lithium-6), but there should not be a great deal of difference between the heavier and lighter isotopes. This is also an argument that most of the post-primordial production of lithium-7 takes places in (giant) stars and novae (see Prantzos 2012).
Boron
Boron is not made in stars - it is actually destroyed in stars. It is also hardly produced at all in primordial nucleosynthesis. The primary source of boron is thought to be reactions whereby CNO nuclei are broken up by energetic protons or alpha particles (Galactic cosmic rays) or the reverse, that energetic CNO nuclei encounter protons and alpha particles in the interstellar medium (the ISM, e.g. Vangioni-Flam & Casse 1999).
The evidence that this occurs is a basic correlation between boron abundances and metallicity - as the numbers of CNO nuclei increase (via stellar nucleosynthesis and recycling), so to does the reaction rate between cosmic rays and the necessary nuclei in the ISM. An early influential paper by Meneguzzi et al. (1971) modelled the interactions of cosmic rays and the ISM. It was very successful at predicting the ratios of boron-11 to boron-10 (though see below) and the ratios of boron to beryllium (which is also not produced in stars or the big bang), but showed that an additional source of lithium-7 was required (see discussion on lithium above).
The basic reason for the boron-11/boron-10 ratio is therefore that the spallation reaction cross-sections favour production of boron-11 to boron-10 by about 2.5:1. The actual ratio in the ISM/meteorites is more like 4:1. For that reason it is thought that there is an additional mechanism to produce boron-11, which is neutrino-induced spallation of carbon nuclei during core-collapse supernovae (Woosley et al. 1990, Prantzos 2012).
Going further down the rabbit-hole, the reason the cross-section for production of boron-11 is higher than boron-10 is apparently because boron-11 is closer than boron-10 to the nuclear stability line and has a higher binding energy per nucleon and this favours its production when carbon nuclei are fragmented (e.g. Webber et al. 2003).
