Where I’m lost is why anyone would be keeping tabs on Tatooine at all. It’s a complete backwater, ignored by both the Republic and the Empire (which is why Jabba has such strong authority there; the galactic governments don’t care enough to take him out), and Vader has completely shed his old life and isn’t looking back. It’s not like he shows up every Thanksgiving to have a chat with Watto or the Lars family. He wasn’t even aware the children had lived. As soon as he had his NOOOO moment, his entire being was centered on carrying out Palpatine’s will.
Also consider: how many people really do know Skywalker became Vader? As far as I can tell the only living people who’d know are Palpatine, Obi-wan, and Yoda. The first has no reason to tell and the other two are in exile. From the time Anakin became Darth Vader to when he was rescued by Palpatine and put in the life support suit, everyone he encountered died except for those three people. An Imperial bureaucrat might notice that this kid on Tatooine has the same name as a former hero of the Republic, but as far as anyone knows he died in the “Jedi uprising.”
Really, putting Luke with the Lars family on Tatooine is no less safe than putting him with any arbitrary family on any arbitrary backwater, and at least they’re blood kin.
ETA: Oh, yes, and Bail Organa, but he has just as much reason to hide the truth as the other three.