Bankruptcy has several levels, for example one simply cancels debt, another liquidates everything, and there is some in between.
I don't think canceling debt is going to help them much.
Infrastructure fees are recurring, as is payroll, what else is there Twitter makes nothing. I'm sure they have bills/ they could vacate but it's probably not enough to matter. It could let him out of repaying any loans taken by Twitter but again, how much is that going to help? If it's not a profitable company it's not a profitable company. As for Musk himself, it won't protect him from repaying the loans he took to buy it and I suspect he had to have used something more than Twitter "stock" to buy Twitter and get a good rate, he didn't use cash, and those banks will come for whatever that collateral was, probably SpaceX and Tesla stock, which he would have to forfeit and pay taxes on.
If that happens, he would lose not only the $12Bil he invested but will be taxed on those stocks when he trades or sells them, about $15Bil, and that's before any lawsuits begin and they've already started. Those could tally more than the 44Bil and if he has to cash out stock to pay it he pays that, plus the taxes, and worse as he does so, the price will crash on those stocks making him liquidate more and more of SpaceX and Tesla as he does so. While I doubt it could go that far he could have just destroyed Tesla and SpaceX. I'm sure he'll come out smelling like roses, he has too much money but those companies may never be the same if that happens.
That however brings up a secondary problem though.
Once a company is in trouble hedge funds will pile on. This was what happened to Gamestop, it became their wipping boy. Need cash, short Gamestop. Price not dropping, spread a rumor and watch it fall. They do this over and over until it just destroys a company then someone like Mitt Romney comes in and buys up the rubble for pennies on the dollar. They will use the media, feds, and blatant lies to cause a stock to tank because if the company folds they never have to pay any of it back, and since they don't own that stock it's not insider trading. It's shady and speaks of questionable morals, but legally it's a bit of a gray area. So if they pile on Tesla and SpaceX, the former of which is well known to be massively over valued and the CEO is cashing out tons of stocks it's only a matter of time before those hedgefunds start circling. Right or wrong, once they smell blood and start to circle it's not easy to recover.
I have no idea how far this will go.
I'm not saying Musk is dumb, he's not, but even smart people need lawyers and a good lawyer, I would think, would have kept him from doing this in the first place, though that may have been what happened when he tried to bail out of the deal. So have no idea what to expect, this is just spitballing end game worst case scenarios and there are many people, and smarter people than me doing it.