Interesting topic, but this stems from something a lot of people seem to be forgetting. Let's check facts first:
The seas were rough already as they were, then Roger roamed free and created the pirate age by talking about One Piece.
Thus, the pirate problem became too big to handle for WG alone, and to balance the power against the yonkous but also to stop the ever rising number or pirates, we have the shichibukai.
So far i think we can all agree.
Now, I don't think WG is stupid enough to ally themselves with their enemy and blindly believe all is good. I'd like to point out that we don't even know what exactly that partnership entails for either side, all we know is this pirates are given a lot of freedom, chase pirates and, during the war, had to fight alongside the marines.
That freedom does have some limits, like trying to take down a country, or allying yourself with a non-WG sanctioned pirate... but that doesn't mean they blindly take orders from WG, as we saw when Kuma talked to Kizaru, as well as Kuma going to help Moria, who kindly refused (why not send marines instead? my guess is if WG did that, those marines would end up as zombies, as WG is not to medle into a Shichibukai's affairs, as Law pointed out in Punk Hazard)
So are they really in breach of their agreement with the WG? We can assume they won't really respect the rules unless it suits them really well, but it doesn't seem like it, even if we fringe on that limit.
Now, let's consider a new problem. The age of pirates started with Roger's talking about One Piece did cause major problems for WG. Now, this problem only deteriorated since we have the worst generation wreaking havoc, and WB blurted out on public den den mushi that One Piece exists. We know WG had their hands so full they had to switch their main base to the New World. This means even with new troops such as Fujitora, they have too much on their plate to deal with everything, and Shichibukais thinning out the herd of new pirates is worth as much as it was when they created that title, even more so.
Also why did they (or as DD said, higher, probably meaning the Tenryuubito) get rid of Moria for being useless, when they don't dismiss Hancock who was detrimental to their army during the war? Was it his loss against Luffy? I don't believe that's all there is to it, since that fight did not break news, Moria was still a sort of tough guy by normal standards (we're used to stronger people now, but that doesn't mean he is not), still instilling fear in common pirates, with his reputation untarnished.
So yeah, as the OP pointed out, they abuse their position as Shichibukai, not having the marines chasing them, to actually scheme and do arguably worse than they could have done otherwise. But for now, i don't think WG can consider getting rid of them, and they have to be able to fully adress the yonkou situation first. This was not the case and it will likely become worse now that Luffy is loose again
Think of Shichibukai as necessary evil, that should be kept under check, but cannot be for two reasons: first, they are pirates and will somehow always elude surveillance, and second they would never agree to cooperate with WG under such conditions. And when they do breach their agreement, WG will try to do something about it, if they can (going against BB after he absorbed WB's fruit would have ended in total destruction so they couldn't, they did go after Jimbei, and they cannot touch DD). They did also chase BB before he got too big to handle by becoming yonkou.
My guess is WB choses to kind of trust these guys just a little, in hope they don't get too much screwed in the end, like with Mihawk, whose partnership looks likely to be more profitable to WG than less, him training Zoro or not. Of course in the end it has to work in favor of our heroes, but we're far from it yet and in that world every character thinks they are the hero, they don't know like us that Luffy is
I could paraphrase stuff that's already been said previously but to what effect if it doesn't bring any new point to this debate. Considering all this, of course we as readers know the Shichibukai are pretty bad news even for WG, but WG has to be at least a little bit aware of it, and chosing to deal with the best of bad situations, and in light of that of course it has to be beneficial to them or at least appear so. I mean, why would they ally themselves to bloody pirates unless it somehow gives them some advantage, for which they are willing to pay such price? There are likely stupid spoiled Tenryuubito like the ones we saw in Shaobondy, but the ones pulling the real strings look more than capable of grasping the situation (you know the ones we saw only once a long time ago, and maybe talking to DD, so we know nothing about them and their motives)
Talking about said motives, it seems tied to the revolutionary army and the true history which Robin is trying to uncover (and likely, Roger's crew knew about) and it looks like staying in control of the world population and beliefs is a good enough tradeoff for letting these pirates roam free, at least for now and until we get more info about all this. I mean between the risk of loosing some power or all of your power, which would a power hungry guy chose?
