Because I think they are doing it to drum up attention for their own manipulative self-interests (IE. Staying in power). Think about the outrage this generated, now they can use it as a rallying point to campaign off of and build support for the next term.
If you want to suggest that the Democrats have leapt straight on SCOTUS's ruling for money and votes, sure. They have. There is certainly a level of cynicism in the Democratic attitude to having abortion encoded in law, but there was also pragmatism: they had abortion rights, and arguably they did have better things to do. Then the Republicans decided to aggressively stack the court and RBG died a few months too early.
IN this video he shows clips of Obama directly saying that signing that abortion bill into LAW was the FIRST thing he would do in office. He lied to make it seem important to him in order to get elected and then for 8 fucking years it was never a priority? Funny how that happens aint it.
You're watching the video of a right wing political activist. He's not presenting facts neutrally.
Obama is specifically referring in that clip to the Freedom of Choice Act, which was doing the rounds at the time. Bills are passed by the legislature not the president, so Congress would need to pass it in order for it to get to his desk in order to sign it. Democratic legislators left it on the shelf, because (so they said) they didn't have the votes to pass it. One could say that Obama could have pushed the legislature to try to get it through, but that was still no guarantee of success and he did have bigger things to spend political capital on (e.g. the Affordable Care Act).
Biden, btw, argued that Roe v. Wade wasn't a solid verdict back in the 1980's. So it's not a new issue for him either. Why isn't he signing the law? Notice how he is calling out the court, when he could issue lawmakers to drum up the Freedom of Choice bill again and sign that shit.
The Freedom of Choice bill
cannot be passed. The Republicans will block it. People like Susan Collins might make moany noises of dissatisfaction at Roe v Wade being reversed, but she is very unlikely to aggravate Republican voters by backing a federal law encoding abortion rights. Even if she and a couple of other Rs did, there are easily 40+ R senators who will block it.
So it requires a Democratic majority in both houses, including a supermajority of 60 in the Senate, unless they can find some way of removing a filibuster. They can't end the filibuster, because some "bipartisan" (ha ha) Democrats like Joe Manchin will shit a brick. (Arguably Joe Manchin might even be right here, because the Republicans look set to dominate the Senate for decades because of all the low population red states, meaning the filibuster is arguably more useful to Democrats than Republicans in the long run strategic view).
Finally, even if a miracle occured and it was passed, consider that SCOTUS has been stacked. Is the Freedom of Choice bill survivable in any form? Because if SCOTUS wants to shoot it down, they can make up a reason to do so. It might be a justification as weak as Roe v. Wade, but it doesn't matter how shit their reason is (they could even literally just say "No, because reasons lol"): once they have killed it, all that effort has gone nowhere and Democrats have to wait for a new SCOTUS composition to go through the whole rigmarole of drumming up a new bill again from scratch.
That's why the Republicans stacked SCOTUS in the first place: the unparallelled ability to decide what laws stick and what don't through the courts. Even if they burn down the reputation of the courts and democracy, they don't give a shit because they've won.