Your conclusion 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:  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 book *Representations of Algebraic Groups* (AMS, 2003), where the results you want are in Part II.          

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.