Ok to clarify only speaking for myself not the org view: (also mostly just theory)
1) Hunting pirates may sound noble....but where is the profit/cost offset? Sounds blunt I know, but as any "theoretical" anti-pirate organization, why would I take on the costs of eradicating pirates, if there is no profit to be made or at least good will and offsetting costs? I mean I am making enemies with a group, and potential friends of that group, that I don't necessarily need to be enemies with.
2) Assuming pirates will be dumb, you have done them a lot of favors. Never underestimate any opponent, these are not actual pirates with no education or reasoning ability but people much like you and I on the other side. If you can "out think", they can too.
3) Anti Pirate means we do not take part in pirate activities nor do we condone them in our ranks. It does not dictate an org's actions in every scenario because "Pirate"!!
4) It is much smarter to patrol controllable trade routes, which when paid for by individuals, they may use them. We secure them, this is a much smaller area than "all of space" or "anywhere in a sector".
It also allows pirate groups an option. If they want to take on the large org with a considerable military presence, they can choose to do so and quickly learn their mistake. Or the easier option, they can prey on those who choose not to pay for and use the secured trade routes, and not get entangled with the large org. Options are good.
5) This in no way means bounties on pirates that are too overzealous will be ignored, in fact they may be taken up by individual groups who feel the need to do good on any certain day, who may or may not be in the org, but not as an ORG activity. In short the pirate needs to be effecting our cash flow for an org to get involved officially.
6) That option mentioned in 4, will keep business as usual. Allow them room to operate, with fixed rules, and everyone is happy. In fact, if an outsider comes in and decides to "pirate it up" in their sector, you may see pirates turn on pirates to keep the status quo, and not get the Large Military presence involved.
All just my view, but it really seems the smarter way to go.
The alternative, of hunting down all pirates, not only seems ridiculously expensive both from a time and cost perspective, it also seems like it could unite them against the big bad overbearing org.