The answer to your question is basically no. Given the vintage of Weil's paper, you can't expect his statement to occur in this form in later books or even lecture notes on algebraic groups. Weil was then working with a language for algebraic geometry which attempted to add precision to what the Italian geometers had done earlier in the century. But his Foundations book did not last long with the coming of other languages, including the hybrid language developed by Chevalley in the 1950s and the more definitive Grothendieck language. (In his lectures and textbook, Borel took some expository shortcuts to avoid scheme language for the most part; but he didn't claim this was an all-purpose way to deal with algebraic groups.)
Unfortunately it takes quite a bit of work nowadays to read the older papers by Weil, Rosenlicht, and others within a modern context, though the effort sometimes pays off.
That said, it would take me a long time to get far enough inside the context of the papers you cite to say anything sensible about what it all means in current language.
UPDATE: An anonymous but usually reliable source (code name X) refers me to Chapter 5 of the 1990 Springer Ergebnisse volume Neron Models by Bosch, Lutkebohmert, and Raynaud. While this book concerns mainly abelian varieties rather than affine algebraic groups, it seems to provide a complete version of Weil's theorem in more modern language. This can be done most simply over a base field but has wider relevance over a Dedekind domain. In any case, X emphasizes that it's unrealistic to work throughout with affine algebraic groups even if that is one area of potential application. (Those with access to MathSciNet may find the review there by Jim Milne helpful, though he is not X.)