The consequences of their sin were distressing to the Israelites. That will always be the case with sin – it never delivers what it promises, and its consequences are often beyond what we expect. Does the prospect of sin separating us from God distress us? If not, these Israelites were in better spiritual condition than we are.
But note that it’s not because of His anger that sin separated Him from His people; it was because He might destroy them in their sin. God isn’t harsh not to accept us sin and all. He can’t be in the presence of sin, and if our choices bring sin into His presence, the result is our destruction. It seems as if destruction is a reflexive response God has to sin. Thus, His refusal to allow us in His presence in our sinful condition is not harsh, but merciful.
He speaks of destroying them, and yet the first manifestation of His glory is His goodness (Exodus 33:19), mercy and compassion. In 34:5-7 the name He proclaims is “the compassionate and gracious God, slow to anger, abounding in love and faithfulness, maintaining love to thousands, and forgiving wickedness, rebellion and sin.” How can we reconcile those two ends of the spectrum characterizing the same person? What does this teach you about God?
God did forgive the people, and their response was to give and serve to fulfill the need for the Tabernacle. His forgiveness should move us to action as well.