The answer to this question is supplied explicitly by Phil 2:5-8:

> **5** Let this mind be in you which was also in Christ Jesus:
> 
> **6** Who, existing in the form of God, did not consider equality with God something to be grasped,
> 
> **7** but **emptied Himself**, taking the form of a servant, made [himself] in human likeness.
> 
> **8** And being found in appearance as a man, He **humbled Himself** and became obedient to death— even death on a cross.

In V7 above, we have three actions:
 - Jesus emptied Himself
 - Jesus actively took the form of a servant
 - Lastly, again, the verb γενόμενος is in the middle voice and thus technically could be translated (as I have above) "made [Himself]".  That is the "making" of Jesus in the likeness of humanity was (according the the verb itself), an act of Jesus Himself

Let there be no doubt that Jesus was the one who emptied Himself and humbled himself and Himself took humanity.  This is confirmed by the middle voice in John 1:14 (as the OP correctly observes) of the verb γίνομαι in the form ἐγένετο = to become by one's own action.  Thus, if I were translating this in an extremely pedantic way, it might look something like:

> John 1:14 - And the Word **became/made Himself** flesh and dwelt among us
> ...

We see further evidence for this in Jesus' famous statement in John 10:17, 18,

> The reason the Father loves Me is that I lay down My life in order to
> take it up again. No one takes it from Me, but **I lay it down of My
> own accord**. I have authority to lay it down and authority to take it
> up again. ...

Further Notes BTW:

1. This is not surprising as in John 1:3 we are told that "all things" were made by the Word. and this obviously includes His own "flesh".  See also Col 1:16.
2. The usual term in English for such "enfleshment" is, "incarnation" (from a Latin root).