This is perhaps a comment rather than an answer but too long for a comment.
Your question points to a general problem that I might call the "unitary model of scientific knowledge", in that there is a singular position in a historical moment that describe what is "discovered", "known", and "rediscovered". The missing question or variable is "to whom?".
As numerous commenters have pointed out, "system theory" never become unpopular or forgotten in engineering. Linear system theory is the core of virtually any engineering program, similarly can be said about other aspects of systems theory such as network theory and it's models (graph theory). This stands in no contradiction at all with some notion of "the mainstream of mathematics researchers show little to no awareness of system theory during a time period" and that in this sense "some mathematics researchers rediscovered systems theory in the guise of categorical systems theory", the the hypothetical at least.
Sadly even on the last statement I consider this ahistorical. Many practicioners in applied category theory have training in engineering disciplines (computer science etc) are very aware of ideas and developments in applied fields such as network theory, so it is in my view not even correct to say that this is a rediscovery. At current categorical systems theory has the status of a category-theoretic reformulation of applied models that have not categorified before.
So the fundamental question here is "what is known to whom" and what constitutes "forgotten" and "rediscovered". The most important step here is to "desingularize" knowledge. What may be unknown to a fresh pure math PhD may be well known to a fresh engineering PhD and vice versa. There may in fact even be differences in knowledge inside these groups and it might further be made more complicated as cohorts of working scientists further mature.
For example, Gabriel Kron provided work in systems theory from the perspective of decompositions of analogue machines, but with the intent of general applicability. I think it is fair to say that much of Kron's work has in substance been forgotten by most engineers and mathematicians, but stuck around as a kind of folklore history (people may know Gabriel Kron as a historical figure). But here too it is important to be precise. Kron's work is much more cited in analogue machine theory today than in other branches of engineering. In digital signal processing some of his ideas have reemerged (Kron reduction) recently. So it is important even on these examples to be specific whose knowledge we are referring to.
