Here is an example relevant to this issue and my work with (the late) Jean-Louis Loday. I visited Strasbourg in November 1981 and gave a seminar on my work with Philip Higgins on a higher van Kampen Theorem. He told me of a conjecture of his on the cofibre of a "connected" square of maps. Looking at this I found that this conjecture could be described as a triadic Hurewicz Theorem. I also explained that in the work Higgins, we showed that the classic relative Hurewicz Theorem could be deduced from our much more general higher van Kampen Theorem for relative homotopy groups. So it would be good to deduce a triadic Hurewicz Theorem from a van Kampen Theorem for triadic homotopy groups. Jean-Louis then was convinced that a van Kampen theorem for his cat-$n$-groups was true, and would be easier to prove than the more special result. This turned out to be the way the work went, and the theorem and this consequence, as well as others, were eventually proved, and they appeared as

R. Brown and J-L. Loday, ``Van Kampen theorems for diagrams of spaces'', Topology 26 (1987) 311-334.

R. Brown and J-L. Loday, ``Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces'', Proc. London Math. Soc. (3) 54 (1987) 176-192.