Ben Stokes would normally bat at No 6 for Trevor Bayliss but England’s vice-captain did not travel Down Under with the rest of the squad following his arrest.
Bairstow was expected to move up one place in the batting order in Stokes’ absence but England instead opted to promote Moeen to six.
England’s batsmen failed to fire in the Ashes opener and Bairstow gave his wicket away cheaply in both innings as he was joined at the crease by tail-enders.
‘I can’t help but think Bairstow is a better player than some of the guys I saw batting above him in the order,’ former Australia captain Ponting told BT Sport.
‘It’s something they might have to look at.
‘Why are they worried about who is shepherding the tail? They should be getting their best batters up there and let them do the batting.’
Swann agreed with Ponting’s point, adding: ‘Moeen has shepherded the tail in England and done a good job – I don’t see why they’ve switched.’
England were hoping to build a considerable lead on day four but were bowled out for just 195, giving Australia a modest chase 170 to go 1-0 in the Ashes.
By the close of play, David Warner and Cameron Bancroft had fired the hosts to 114-0 as they closed in on a crushing victory.
‘England have had a poor day,’ ex-England skipper Michael Vaughan said. ‘A lot of the fears before the series have come to fruition.
“Moeen Ali has been half the bowler Nathan Lyon has.”
— The Ashes on BT Sport (@btsportcricket) November 26, 2017
‘Australia bowled with such hostility, you have to give them credit.
‘The problem for me is that there were chinks in Australia during this match but within a day and a half we’re looking at a lot of damage that has been done to England. They’ve been blown away once again.’