Ok, still Summers would have voted for BBB when Manchin wouldn't. He says as much in one of those quotes:
"No, Summers said BBB should be fully paid for to not add to inflation...and more.. ."
"Summers stated, “Look, my view is that, on balance, even if you had it as it stands, Build Back Better is better
legislation for the United States. If I were in the Senate, I would vote for the Build Back Better legislation."
So he and Manchin were seriously at odds rather than as close as some leaning to Manchin would suggest. I'd guess the WH had inflation concerns about the bill too, but felt, as it seems Summers did, better it go through regardless. Putin's war, as well as the sheer cruelty in itself, with it's exacerbation of the covid supply chains problems must have really stung the supporters of the bill.
Again, obviously each side bears some responsibility for the bill failing. Still according to that quote it reads to
me if Summers had been in Manchin's place BBB would have passed. Is that a fair reading of that quote?