That obviously wouldn't work. There aren't many measures that "the Squad" would have made a difference on - and the few bills they would have been able to draw a line on were vital votes for other services and issues. They're getting demonized right now for pushing Pelosi and Schumer a little bit harder on the infrastructure bill, there is little chance that they would have been able to stop anything important without getting destroyed, nationwide.
Not only that, but you're pointing out that Congress was aware of the deadline - the administrative agencies, the federal courts, the executive branch, and all three branches on the state level(s) were also aware of the deadline, and each assumed that the other would pick up the slack. I'm not going to attack the few legitimately progressive Congresspeople for making a national point of protest when the failure is evenly spread along every branch of government in the nation.
There's also the point that no one in this thread seems to understand: the progressive wing already tried to get the moratorium extended - the moderate and conservative wing of the party explicitly opposed it, alongside the Republicans - because the Biden Administration purposely didn't make its position known until July 29th.
How Democrats Screwed Up So Royally on Evictions