What possible motivation would Qatar and the others have for that?

To get rid of Hamas. Or prevent the PFLP from getting a foothold in Gaza.

Remember, Hamas is the Gaza offshoot of the Muslim Brotherhood - and Egypt absolutely wants to crush the last lingering remnants of Hamas for that reason alone. BTW, that's why the "open-air prison" that international rights groups have condemned doesn't have a huge wide-open gate on the southern end - Egypt absolutely didn't want the Palestinians to have free movement or economic interaction with Egypt in order to keep Hamas out. And Jordan is violently opposed to the militant factions within the PLO/PA like PFLP.

As I mentioned upthread, you can kill (or at least really weaken) an idea - and one tool for doing that is brutal repression of anyone who tries to promote that idea. The new "Board of Peace" may be highly motivated to prevent any of the militant organizations from having even the slightest hint of a glimmer of the possibility of a foothold in Gaza. Because a power vacuum invites people to try to seize power, and the militant groups in the region will assuredly try to get a position in Gaza if they're not squelched. And if there's one thing these regimes in the area are good at, it's squelching.

What does more chaos and misery on their border buy them?

That's the point. They don't want chaos and misery, so they're going to choose the order of brutal repression of dissent and any political activity that they don't like. Freedom and self-determination is usually far more unruly and chaotic than an authoritarian regime - and it's a near certainty when some of the ideas that are jockeying for acceptance involve militant resistance and violent overthrow of the established governments in the region. Which is why if the BoP countries actually want to keep control of Gaza, they're more likely to take the "repressive authoritarian" route than the "lets give the Palestinians freedom and autonomy" route.