So again you need to look at the context of the scripture ----
Matthew 10:16
A Hard Road before Them
16 “Listen carefully: I am sending you out like sheep among wolves; so be
wise as serpents, and
innocent as doves [have no self-serving agenda].
So here he is sending out His Disciples into the world of unbelievers ------a pack of wolves -----
So for me this is saying spiritually -----that Jesus is telling His Disciples to be like the serpent who displays cunning and shrewdness behaviour ----
So Jesus is using the serpent symbolically here to caution them to be weary of the danger and conflicts that are going to arise as they proclaim God's word ----and not to loose the Dove Like character ---which is peace ---purity and gentleness that will allow them to Proclaim God's word without fear -----so they have to find that balance that can keep them safe but yet keep their Christ like Character in tact --
Genesis 3, the narrative does not suggest that the talking snake is Satan..it appears the two concepts are merged along with the later Hellenistic period
You are right here ---it does not mention Satan's name with the serpent here but what is does do --it describes Satan's characteristic which is crafty and shrewd ----
From Bible Hub -----
Lexicon
arum: Shrewd, crafty,, cunning
adjective crafty, shrewd —
crafty, ׳ע as predicate, of serpent
Genesis 3:1
Genesis 3:1 --
Amplified Bible
Now the serpent was more
crafty (subtle, skilled in deceit) than any living creature of the field which the LORD God had made. And the serpent (Satan) said to the woman, “Can it really be that God has said, ‘You shall not eat from any tree of the garden’?”
then Revelation 12:9 ---says this which does really clarify Genesis 3:1
AMPC
And the huge dragon was cast down
and out—
that age-old serpent, who is called the Devil and Satan, he who is the seducer (deceiver) of all humanity the world over; he was forced out
and down to the earth, and his angels were flung out along with him.