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 $L(0)$ at the top, but the linked module $L(2p-2)$ 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.