I mean Republicans are why these programs persisted. Why things remain so fucked. Why immigration reform hasn't happened. To single out Obama and almost absolve Republicans as if they were merely bystanders is a bit of a misrepresentation of history. If it were up to most blue states, Democrats, and eventually the Obama administration onward, we would of had immigration reform in 2006, 2010, and 2013. None happened because Republicans refused to allow a vote in the House in 06 and 13, and killed the other in 10. And yes I am sure someone will say why didn't they do it in 08-10, and to an extent that is valid, but it is also a bit unrealistic given the world the Democrats inherited. By the time Kennedy died any chance of an immigration bill was dead in the senate and Pelosi I think fairly made the calculus that taking it up in the house was fruitless if they couldn't get it through the senate. Which they couldn't. Which was her rationale at the time.
The problem with that series of tweets, after reading about it more, is that it is conflating two separate polices/issues. One that did occur under the Obama administration that when parents caught by ICE for being here illegally, often in red border states, that had US born children, they were separated and put in the care of relatives or in some cases foster homes when that was not possible, while the parents deported. Which is a separate policy/issue from what the Trump administration is engaging in which is seperating families at the border and straight up losing track of children. I'm not a fan of either policy but it is a bit misleading to infer that it is just a straight continuation.