As indicated in the comments, there is no need to redo the entire classification argument when passing from "semisimple" to "reductive". But it's useful to recall some of the history. The emphasis in the Chevalley seminar 1956-58 was on achieving a uniform classification of semisimple algebraic groups over an algebraically closed field of arbitrary characteristic. For this a slightly more general treatment of isogenies between such groups is more natural. (Chevalley discovered for example that the simple groups of types $B_\ell$ and $C_\ell$ are isogenous in characteristic 2 while their groups of rational points are isomorphic even though the underlying algebraic groups are not).
A little later, Demazure and Grothendieck translated most of this into the more flexible language of group schemes in SGA3, while Borel and Tits by 1965 expanded the framework by defining reductive algebraic groups over an arbitrary field. (For such groups Tits achieved a classification, modulo some later reformulation of his theorem stated in the proceedings of the 1965 Boulder AMS Institute. The main unsolved problem is to classify the $k$-anisotropic groups, which vary a lot for different fields of definition $k$.)
The Chevalley seminar and the other sources in French are available online from numdam, but note that SGA3 has been
re-edited in recent years while a corrected typeset version of the Chevalley seminar was published in 2005 by Springer. (In my 1975 textbook, I mostly followed the Chevalley seminar; but when the characteristic is not 2 or 3, my 1966 thesis showed that it's also possible to rely more on the Lie algebra as in the classical characteristic 0 case.)
Variants were found by M. Takeuchi and T.A. Springer, using for example ideas of Serre and Steinberg about generators and relations for the groups. But in spite of the differences in the published approaches, all require a lot of detail about the internal structure of simple algebraic groups (those with no proper closed normal subgroups):
Bruhat decomposition, generation by tori along with root subgroups.
A key conclusion is that two such algebraic groups are isomorphic precisely when they have the same root system (or Dynkin diagram) and the same fundamental group, except in type $D_\ell$ with $\ell >2$ even, when you have to distinguish the half-spin and special orthogonal groups.
(Note too that the work of Tits gave a more precise picture of the internal structure: when $G$ is a simple algebraic group, its only proper normal subgroups are those contained in the finite center.)
Using the methods of Chevalley (1955), one further shows the existence of all possible types of simple algebraic groups. The classification of possible semisimple groups is then a routine but slightly messy exercise: start with a product of simply connected simple groups, then factor out a subgroup of the (finite) center. There may be a great many possibilities.
Translating all of this into the language of reductive groups is then a matter of reading between the lines in Springer's textbook: given an isomorphism of root data, one gets an isomorphism of root systems along with a comparison of fundamental groups, etc. Unfortunately, there is no single source in the literature for a truly unified treatment of all these matters, including isogenies. But the core of it all is the study of the Borel/Chevalley structure theory. The transition to reductive groups is needed mainly because these are more natural for induction purposes than the semisimple ones: Levi subgroups of parabolics are reductive but seldom semisimple. However, central tori over an algebraically closed field are fairly innocuous.
