ougj this really reminds me of the fact that most fics I've read had him completely alone out there. Which, is extremely fair if you're taking the very probable situation of Keith bouncing around foster houses for a while (I forgot what happened throughout those years in canon, don't come at me)). Nobody really knew he was living out in some bullshit shack that belonged to his father, which makes sense seeing that the only person he could have mentioned it to would be Shiro.
Through that, Adam may have learned about it, and if he remembered, he might have guessed that's where Keith would go after getting kicked out. Seeing as he had nowhere else.
Relying on the version of events where Keith and Shiro were exceptionally and truly close back in the Garrison, this could have happened.
But I also present you all with: Adam meant to reach out. He really did. He'd been considering sending something but everything was so busy and Keith was too closely tied to a Shiro he had just lost. Keith had always been independent to a fault, and he might see a letter of concern or money/supplies sent as almost an insult. Insinuating that he couldn't make it on his own.
Eventually Adam braves it. Eventually, he's too guilty to put it off any longer, and thinks to go check for himself if the kid's even alive. It takes him some time to find the shack, old and creaky. And by then, it's much too late.
š (Keith would have been grateful to be sent things. Keith would have been honored to be remembered, a part of him had been waiting for it the whole time. He lived with the knowledge any one of the people he knew could have reached out, at any point.))