Well, Outsiders as a group don't seem to have a weakness the way that the Black Court does. This makes it really difficult to out them to the world in order to take them down.
Then of course there's the fact that the things in Lovecraft's stories are rarely defeated. A witch here or a wizard there might be killed, but the horrible things from beyond the Earth are rarely even inconvenienced. Most of them barely seem to notice that humans are present at all.
Combine these two facts, and it seems exceedingly unlikely that Lovecraft was trying to out the Outsiders. If he was, he did a genuinely awful job of it. As a method of instruction, his stories were mostly how to die when something terrible decides to kill you. Thanks, Howard. I think I could have managed that myself.
I think that the path of least resistance goes one of 3 ways:
1. He was an author who wrote fiction. If he managed to hit any truths related to the Outsiders, it's more or less happenstance.
2. Either on his own or at the behest of someone else, Mr. Lovecraft was trying to put information out there as part of a plan to help one or more Outsiders enter the world. Whether this was successful or not is up in the air.
3. He was trying to warn people: Mess with this stuff, and it will go very poorly for you.