As far as Vader knew the baby died with Padme. As for not recognizing Leia, no one even knew Padme was having twins until she was giving birth to them. Momentarily setting aside that absurdity, because in a world of such amazing technology there is no way the doctors would not see two heartbeats. Then again the delivery Droid had spatulas for hands and spoke in creepy robotic baby talk (droid literally says ooooobahhhhh like some dyslexic half wit) and Padme died of a broken heart so maybe it's not that far fetched. Man ROTS is a mess! That aside, Vader would have no reason to go looking for Luke. Thus it makes logical sense to hide Luke on Tatooinne which was a nothing planet. Plus Anakin hates sand. 'Nuff said. A whole planet of that stuff? No thanks bro!
I would find it problematic to have Obi-Wan go into hiding to protect the knowledge of Luke's existence from Vader and the Emperor, only to go on covert missions off world. It could very well undermine the nobility of Obi-Wan as a character because he swore an oath to protect the child of his greatest failure. Luke is literally the personification of Obi's redemption.
Not only does this act speak to Obi-Wan's humility to give up his life to watch over the child, but because he was willing to live on a desolate world to do it. He could have easily found some resort planet and lived in luxury but he chose not to.
Going off world on a myriad of adventures not only risks Luke's safety but it also undermines Luke's importance being A New Hope. I guess you could argue that their location could be compromised and Obi-Wan has to leave sometimes to protect its secrecy, but it seems unlikely and would beg the question why he couldn't just grab Luke and run for it if their location was compromised.
Also taking into account that the Jedi and Obi-Wan especially are fugitives of the Empire and have few allies left so showing up to save the day off world means unnecessarily putting a target on his back and by extension Luke, thus possibly defeating the entire purpose of going into hiding in the first place. The whole point is staying under the radar and leaving only risks calling attention to yourself. As if Obi-Wan needs any more reason to be humbled.
I know the likelihood of my next suggestion is too cerebral (at least for the masses) and I'm putting in in Spolier tags to avoid the risk of getting swarmed by the apologist. Granted this idea would take a very skilled writer to pull off so the chances of it happening is almost zero but.....
You could do what The Last Jedi tried and failed to do with Luke, which is a smaller scale character study delving into Obi-Wan dealing with his failures, trying to come to terms with the mistakes he made and furthering his understanding of the Force and how he can fix everything and occasionally going on local adventures on Tatooinne. This would deepen the character and give us insight into the Force, coming to terms with guilt, wrestling with faith and doubt, learning about responsibility and humility. It would be a phenomenal acting opportunity for Ewan who is more than talented enough to give it real emotional weight. But I doubt there are any writers willing to take the risk, or even talented enough to attempt it. It will likely just be more of the same and a chance to have scenes with lightsabers and insane Force abilities.
This really needs to be a limited series too. The only way this show works in any capacity is if they show some restraint. I was fooled once with the first season of The Mandalorian, but we all know how it turned into The Gondolorian, so I have every expectation we will be given more of the same.
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), or just something that caused Obi Wan to reveal himself, then that would require him to chase that individual down. They wouldn't necessarily require the secret of Luke to be known, just that there's a surviving Jedi hiding out on Tattooine. If that gets to the Inquisitors, who report to Vader, then that'll being undue attention to the planet that could end up revealing Luke.
