in principle, the israeli state should not be shutting down any media organizations. the unquestionable factual reality that al jazeera is a horribly biased, openly anti-israel source that spews factually inaccurate, skewed, brutally anti-jewish propaganda out to the west, and that their warped framing of the conflict as "israel's war" is actually having some effect on a subset of the population, is a reason to not shut al jazeera down.
however, al jazeera is state media run by the qatari sheikhs. it's a mouthpiece for the qatari government, who has played a major role in representing hamas. where does the head tyrant of qatar actually live? in a cushy villa in qatar, where he's bankrolled by iranian nazis. qatar is also an iranian beachhead on the arabian peninsula. when one of the primary news sources, as poor and propagandistic as it is, is also a lead negotiator, that creates a conflict of interest. eliminating that conflict of interest should help focus the qataris more on the plot.
that doesn't mean it's something that should be supported, exactly, but the bbc is wrong in it's analysis, at the least.