I am using a 2SLS, with two endogenous variables and two instruments. I conduct the Under- and the Weak-identification tests; results from these tests suggest that my instruments are not weak. Moreover, in the two first stages (i.e. regression of the instrument on the endogenous variable), the instruments have always a statistically significant effect on the endogenous variable. Finally, although I cannot conduct that over-identification test (i.e. I have as many endogenous variables as instruments), my arguments on the validity of the two instruments are solid (grounded on the literature).

In the second stage, the two predicted variables from the first stage have a statistically significant effect on the outcome.

It seems like things are working just fine. However, my question is this:

In the reduced form (i.e. regression of the two instruments on the outcome variable), one of the two instruments does not have a statistically significant effect on the outcome variable. What does that imply?

It seems like displaying the results from the reduced form of the 2SLS might be needed to show that you are not "inventing" the results (my own interpretation of the third answer here) and thus just to reassure the referee (in the case you are working to revise a paper submitted to a journal). I cannot evaluate this answer, but might it hide some truth? (i.e. there might be no technical reason nowadays)

[I have read somewhere that the reduced form might be used as an additional indirect way to test whether the instruments are weak; see section 11 of this document or see this short article. However, I am not sure I get it]