I'm not sure this is an answer, but it got too long to be a comment! The projectivity of $G/B$ seems to me to follow from two facts: a) a homogeneous space $G/H$ is always a quasi-projective variety (which is due to Chevalley), and b) the variety $G/B$ must be complete.
The first fact clearly has nothing to do with Borel subgroups but it would be maybe interesting to ask about how many proofs we have of it. The idea of the construction is clear: find a representation of $G$ which contains a line $L$ which has $H$ as its stabilizer and then take the orbit of $L$ in $\mathbb P(V)$, but to see that you get a categorical quotient this way takes some more care: you need to use some infinitesimal properties, (which you can tidily say using the Lie algebra of course).
The second fact perhaps depends on what you're willing to assume, but it must come down to the Borel fixed point theorem in some form or other, since this tells you that if $G/H$ is complete, then any solvable subgroup will be contained in a conjugate of $H$, thus Borel subgroups are the only solvable subgroups with a chance of having an associated homogeneous space which is complete.
The argument from Humphreys' book uses the strategy of picking first a maximal solvable subgroup of largest dimension so as to show the orbit has to be closed (essentially because the stabilizer is largest so the orbit has smallest, but even then you use the Borel fixed point and the theorem for $GL_n$ if I remember correctly). I don't know how Onishchik and Vinberg get around this (if they do). Then, as the question says, you get the general result from the Borel fixed point theorem.
The proof for $GL_n$ shows that $G/B$ is projective if you take $B$ to be the subgroup of upper triangular matrices, but one still needs to show that this subgroup is a Borel, which again I only know how to do using the Borel fixed point theorem in some form. (And of course you can use the same strategy for other classical groups if you can eyeball a candidate Borel subgroup).
Given that, it's maybe worth pointing out that in all of this you can get away with a weak version of the Borel fixed point theorem: namely if $V$ is a representation of a solvable group $H$, and $X$ is an $H$-stable closed subvariety of $\mathbb P(V)$ then $X$ has an $H$-fixed point. This can be shown just using the Lie-Kolchin theorem (that is, that a representation of a solvable group contains a one-dimensional subrepresentation), which can be proved directly. Since the standard proof of the general Borel fixed point requires you to use something like Zariski's main theorem, this is maybe a noticeable saving.
All of this leads me to wonde how many proofs do we know for i) the fact that for any closed subgroup $H$ the homogeneous space $G/H$ has to be quasi-projective and ii) the Borel fixed point theorem (or some variant)?