It's really not that complicated. They were trying to win, then Whitlock blew the save in the 8th. They went with Barnes next, and got a scoreless inning from him, but if they didn't score in the 9th, they were going to have to use someone not good. There were no good or even decent options left. Anyone they bring in at that point looks like a questionable move.
As for the DFA, someone was going to be dumped when Yu Chang was activated today, so they gave Familia one last chance to see if he could do anything positive (and possibly because they thought his splitter might work against the batters they had coming up.) He was brought in to face the bottom of the order, but blew his last-gasp chance to save his job by sucking again, and so he is the one gone.
The other choices were Ort, Kelly, Danish, Strahm, or Bazardo. Strahm is a lefty, Ort has a 6.33 ERA, Kelly has thrown 6 innings in his major league career, Bazardo has thrown 8 innings in the majors (and has a 7.20 ERA). I would guess that Cora would prefer not to use one of the rookies (old though they may be, they are still inexperienced at this level) in a high-leverage spot against the Yankees-- the rage that occurs from the home fans when we lose to NYY might not be something he thinks they are ready to handle. (Then again, Whitlock probably won't be available tonight, so one of them might have to learn how to handle that anyway, we'll see.)
That leaves Tyler Danish, who at least has 46 major league innings in his career. He was probably coming in next if we tied it up.
So everyone is in a rage because they went with Familia over the great Tyler Danish.
In the game thread, everyone wanted Familia DFA'd as soon as possible. So they DFA'd him as soon as possible, and now people are like "What?!? They DFA'd him? That doesn't make any sense!"