Your conclusion about direct sums is false.  It's helpful here to have some examples in mind, such as a typical projective/injective module for the Lie algebra of $G:=\mathrm{SL}_2$, lifted to $G$. This is *indecomposable* but might have a Weyl filtration with two quotients:  for example, the trivial 1-dimensional module $V(0) =L(0)$ at the top, but the linked Weyl module $V(2p-2)$ at the bottom.  (Here I am referring to a non-negative integral multiple of the single fundamental weight just by the integer coefficient, while $L$ as in Jantzen denotes a simple module.)

In any case, you seem to be referring to the expanded second edition of Jantzen's 1987 Academic Press book *Representations of Algebraic Groups* (AMS, 2003), where the results you want are in Part II.    For example, he refers to 4.19 as II.4.19 if the part is unclear from the context.        

Note too that Jantzen's treatment is for a connected *semisimple* algebraic group, though adaptations to a reductive group with nontrivial (semisimple) derived subgroup can be deduced easily.   And the field of definition doesn't play a role here.