That's fair, they didn't prove an "arsenic backbone," but here's a couple of key lines...

"...the purified DNA extracted from +As/-P cells contained As. Our NanoSIMS analyses, combined with the evidence for intracellular arsenic by ICP-MS and our radiolabeled AsO4- experiments demonstrated that intracellular AsO4 was incorporated into key biomolecules, specifically DNA."

and

"...the level of PO4 impurities in the medium was insufficient to elicit growth in the control (-As/-P)"

I agree. Note that the cells are not phosphorus-free. The paper states that the cells grown without phosphorus contained 0.02% phosphorus, presumably due to impurities in the reagents. Based on the numbers in the paper, it seems like this is enough phosphorus for one copy of the DNA genome.

My hunch is that the bacteria are able to use this low level of phosphorus, rather than replacing phosphorus with arsenic wholesale. It seems unlikely to me that the bacteria have evolved to use arsenic in place of phosphorus in the DNA, in ATP, in phospholipids, and in everything else. It seems even more unlikely that they can dynamically switch between arsenic and phosphorus for all their purposes, since all the enzymes would need to work with two different substrates. (In other words, changing an enzyme to work with As instead of P seems hard; changing an enzyme to work with As or P as needed seems much harder.)

That said, I think this is a cool discovery, and it will be interesting to see just how much of the biochemistry is actually using arsenic, since it seems that at least some is.

They calculated that there's definitely not enough P in the bacterium to sustain all the processes it's running with regular DNA.

..based on their MS measurements. They can't accurately calculate for impurities that might be in the source medium, there's no negative control on that.