Re: John 5:19-21. After Jesus healed the lame man at the pool of Bethesda and the Jews wanted his head for claiming he was God, He said that He did nothing of Himself but only what the Father does. It appears in these verses in John 5 that the Father was doing independent work of His own. So even though He equipped Jesus to do the work while on earth, the Father was also busy working Himself?
I think the answer you’re looking for is in the previous few verses. John 5:16-19 says,
So, because Jesus was doing these things on the Sabbath, the Jewish leaders began to persecute him. In his defense Jesus said to them, “My Father is always at his work to this very day, and I too am working.” 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.”
Some people think God never did anything after the 6th day of creation. But both the Old and New Testaments give evidence to the fact that it was only the work of creating that stopped. Jesus confirmed that God has constantly been at work ever since.