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My question is, given that the Robertson–Seymour theorem to all (undirected) graphs, why did Friedman choose to restrict this function to subcubic graphs?

Short answer: For subcubic graphs, it has been proved that the condition $(|G_i|\le i+k)$ ensures the existence of a longest sequence of the kind being considered. As far as I know, similar conditions haven't (yet) been proved sufficient for that purpose in more-general cases.

Longer answer: The Robertson–Seymour theorem states that finite undirected graphs, quasi-ordered by the graph minor relation, form a well-quasi-ordering; i.e., there is no infinite sequence of such graphs that avoids having some element being a minor of some other element. Thus, every sequence that avoids such an embedding must be finite.

However, without further restrictions, such emdedding-free finite sequences can be arbitrarily long. Ensuring the existence of a maximum attainable length of such sequences is the purpose of the additional condition $(|G_i|\le i+k)$, which imposes a maximum size on each graph in the sequence -- but only for certain kinds of graph has it has been proved that this condition (or a similar condition) is indeed sufficient for this purpose.

Friedman showed the sufficiency of this condition for subcubic graphs (both simple and non-simple); so, for such kinds of graph, it is known that there exists a longest sequence $G_1,G_2,\dots,G_n$ such that $|G_i|\le i+k$ and no $G_i$ is a minor of $G_j$, where $i < j$.

An easier-to-understand special case of this is Higman's lemma applied to sequences of finite words on a finite alphabet. Every sequence that avoids having some element a subsequence of some other element must be finite, but such sequences can be arbitrarily long. However, if we impose the additional condition that the $i$th word in the sequence have length at most $i$, then it can be shown that there exists a maximum attainable length of such embedding-free sequences. (This is the basis of Friedman's "Block subsequence theorem". See also Higman's game.)