God's wrath is certainly directed at sinners.'God is angry with sinners every day (Psalms 7:11). 'And the LORD has laid on Him the iniquity of us all' (Isaiah 53:6). 'Yet it pleased the LORD to bruise Him; He has put Him to grief' (Isaiah 53:10). Surely even you can compare Scripture with Scripture and work that out. God anger against sin and sinners was (willingly) borne by Christ. Christ also suffered the curse of God against sin (Galatians 3:10-13). God also set Him forth as a propitiation, that is a sacrifice that turns away wrath (Romans 3:25-26). .
The word 'appease' is one that I do not care to use regardless of its dictionary meaning because of its unfortunate association with the appeasement of Hitler.. What Christ did was to provide satisfaction for the outraged justice of God, so that He may be 'just and the justifier of the one who believes in Jesus.'
Isaiah 53:5.
And yes, Christ shared in our infirmaty. He bore our sins in His body. And yes, it was God's will that Christ suffer and die by the evil men of this world. And yes, God is Just and the justifier of sinners. And yes, it is an abomination to God to condemn the Righteous and acquit the guilty.
But no, Penal Substitution Theory is not biblical.