The Jews did not make a false pretense in John 10:33, and there's nothing in the context to suggest that is the case. How many times in the discourse does Jesus call God, "Father"? Are you suggesting that the Jews stood around with their fingers in their ears? The Jews heard Jesus loud and clear, and knew what Jesus was claiming by calling God His own Father. Unfortunately, you miss the point.
Pay particular attention to v. 36. Because Jesus actually reiterates and clarifies what the Jews are accusing Him of in v. 33 — “
you say… ‘You are blaspheming,’
because I said, ‘I am the Son of God.’” Jesus is not correcting the Jews, rather, He restates what the Jews are already accusing Him of: blasphemy for claiming to be “the Son of God,” which is equivocal with “making Himself equal with God” (cf. John 5:18, 19:7). So it’s not that the Jews understood Jesus “incorrectly” in v. 33.
In John 19:7, the Jews indicate that Jesus claimed to be “the Son of God” and for this reason, He deserved death. This alone should tell us about Jesus’ very own application of such title — “the Son of God.” But more to the point, this coincides with what is said earlier in John, as there is a direct correlation between the words spoken in John 19:7 and those in John 5:18 (“For this reason they tried all the more to kill Him; not only was He breaking the Sabbath, but He was even calling God His own Father, making Himself equal with God”), and John 10:25-33.
Jesus’ application of the title “Son of God” carries with it its fair share of nuances, but might I suggest that one of the more prominent nuances in NT usage is an extremely Jewish one? What I mean by that, is that the “Son of God” epithet is used frequently of the Davidic King, who is God’s vicar, God’s “right hand man,” who mediates God’s presence, and is in that sense (by way of extension), “equal with God.” But I also understand that Jesus’ application of such title runs even deeper than that of the Davidic King motif, for even the Jews of Jesus’ day understood Jesus’ application (what I would consider a more personal application), as going beyond the scope of what any man could rightfully claim for themselves without the charge of blasphemy being brought against them (hence the, “you being a man make yourself out to be” tid bit).
In both, John 5 and John 10, Jesus acts as God acts — in inseparable union — “For this reason the Jews were persecuting Jesus, because He was doing these things on the Sabbath. But He answered them, ‘My Father is working until now, and I Myself am working.’” Jesus uses the present middle indicative for the word “work,” which indicates a presence of past action still in progress. Just as the sustainer of all things (God the Father) continues to work (throughout history) and is thereby exempt from the rules of the Sabbath, in this same manner, Jesus too has been working (hence, 5:18). Jesus’ works are co-extensive with the Father’s. Thusly, Jesus’ application of “the Son of God” epithet of Himself, is intrinsically tied (in a facet of ways) to the idea of Him being “one with” and “equal to,” God. And as in John 5, it is this application in John 10 that leads His Jewish audience to react in such manner.
Given the echoes of Ps. 95 and Deut. 32 in the John 10 discourse, might I suggest that the anarthrous θεὸν in v. 33 is probably best understood — by no means as indefinite — but as qualitative (i.e., “Who died and made you God?” or “Who made you
[equal with] God?”), and therefore understood as a reference to Jesus’ equality with God (as in John 5:18), as Jesus claims to do the very prerogatives of God (as he does in John 5);
“My sheep hear my voice, and they follow Me; and I give eternal life to them, and they will never perish; and no one will snatch them out of My hand...' For a good work we do not stone You, but for blasphemy; and because You, being a man, make Yourself out to be [equal with] God" (John 10:25–33)
The words that Jesus utters in vv. 27–28 — “My sheep hear My voice, and I know them, and they follow Me; and I give eternal life to them, and they will never perish; and no one will snatch them out of My hand” — are of particular interest. For not only do these words echo that of Psalm 95, but the contextual setting in which they apply run parallel to one another,
For He is our God,
And we are the people of His pasture and the sheep of His hand.
Today, if you would hear His voice,
Do not harden your hearts, as at Meribah,
As in the day of Massah in the wilderness,
“When your fathers tested Me,
They tried Me, though they had seen My work.
“For forty years I loathed that generation,
And said they are a people who err in their heart,
And they do not know My ways.
“Therefore I swore in My anger,
Truly they shall not enter into My rest.” (Psalm 95:7-11, NASB)
Make the connections: John 10:27 (“My sheep hear My voice”) parallels Ps. 95:7 (“Today, if you would hear His voice”). John 10:28 (“My sheep… no one will snatch them out of My hand”) also parallels Ps. 95:7 (“we are the people of His pasture and the sheep of His hand”). John 10:28 also parallels Deuteronomy 32:39 (“there is none that can deliver out of My hand”) and Isaiah 43:13 (“none can deliver from My hand”). John 10:32 (“I showed you many good works”) parallels Ps. 95:9 (“They tried Me, though they had seen My work”).