The fundamental answer is that very little lithium is produced in the big-bang and then nucleosynthesis routes to produce lithium in stars (which is how all the other abundant elements are produced) require temperatures that are far higher than the fusion reaction that readily destroys lithium inside stars.
Lithium is a scarce element in the universe and the abundance of Li in the Solar System and in the Earth's crust is low compared with elements like carbon, oxygen, silicon and iron.
The Solar System lithium is created partly (only 10%) by primordial nucleosynthesis, a bit by spallation reactions of cosmic rays on nuclei in the interstellar medium, but mainly in the interior of relatively low-mass asymptotic giant branch (AGB) stars and in nova outbursts (e.g. Prantzos 2012). The main reaction mechanism is the fusion of helium-4 and helium-3 to produce beryllium-7. This then undergoes electron capture to lithium-7.
Whilst there is plenty of helium-4 inside stars there really isn't much helium-3, except where it is produced in hydrogen-burning cores/shells, but these regions are also hot enough to quickly destroy lithium-7 through proton capture back to helium-4 nuclei. Thus one needs special conditions where Be-rich material from the core/shell is mixed upwards and undergoes electron capture in regions cool enough for the Li to survive (Cameron & Fowler 1971). This can happen in "hot bottom burning" AGB stars with masses of about $4<M/M_{\odot}<8$, which are undergoing shell H- and He-burning for some of the time (e.g. Garcia-Hernandez et al. 2013). The convective envelope reaches down to the H-burning shell, dredges up Be-rich material, which then becomes Li-7. The process is of limited efficiency, since the same convection takes a lot of the Li-7 back down again to be burned. So, although AGB stars can efficiently blow enriched material into space through their massive winds, the material isn't that enriched with Li.
The Cameron & Fowler mechanism can also take place in novae explosions that occur when matter is transferred from a companion onto the surface of a white dwarf and detonates. The accreted material needs to have helium-3 in it, so must also have come from regions where there has been incomplete hydrogen burning. Fast, explosive ejection of a Be-rich shell then results in enrichment of the ISM with Li-7. It turns out that the special conditions required to accrete material with lots of He-3 do not result in enough Li production to boost the interstellar medium Li abundances beyond what we see.
But I think the main thrust of the question is why isn't Li just produced from some sort of fusion reaction, like helium or carbon?
The answer is that it is! For instance Li-7 is produced as part of the PPII branch of the pp chain, at temperatures between $1.4\times10^7$ K and $2.3\times 10^{7}$ K. But at these temperatures the Li-7 is rapidly fused with a proton to form two He-4 nuclei.
So the basic problem is that in stellar interiors, Li-7 is readily burned at temperatures above $3\times 10^{6}$ K, but any fusion reactions that produce Li (or elements heavier than Li) require much higher temperatures than this.