I can find no evidence that Michael was the son of Elizabeth Mitchell who arrived on the Lord Wellington in 1820. I also can find no evidence that he was one of the castaways from the Active. If the dates for this incident are correct he would have been a young child. Youngsters of 10 and up were accepted into the Royal Navy but I am not certain that they were taken on as crew for sealing expeditions. The earliest record for him in Australia seems to be his arrival on the Lord Wellington in 1819. There is another record for a Michael Mitchell arriving on the same ship in 1816 and assigned to William Grono at Windsor. I don't know about the significance of this entry if any to 1820 Michael.
Elizabeth's children who arrived with her are listed in the 1822, 1825 and 1828 musters and census. Michael is not listed in any of these as being a child of Elizabeth. After closing off from the Lord Wellington Michael signed on with the crew of the 'Lynx and then with John Grono as master on the Brig 'Elizabeth' He made several voyages on this Brig sometimes with John Grono as Master and sometimes with Alexander Books as master.