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[Star Wars] Had Anakin Skywalker never been burned and used the suit, would Luke learn of his identity much earlier?

So after the Obi-Wan and Anakin battle, Anakin is left burned and near limbless. This prompted for him to have to use the iconic Darth Vader respirator suit. 

Luke eventually learns of his heritage later on...and the rest is history. 

However... what if things went differently??

**SCENARIO:** Obi-Wan manages to knock Anakin out and leaves on the ship. He drops the babies off to their respective future homes. Anakin wakes up, somewhat humbled but furious. But he continues his path , secretly, as Darth Vader.

Anakin now does not need to hide his appearance and keeps his full name for the public. 

Luke grows up and notices this man on the news who he looks strikingly similar to, especially in the eyes and hair. Not to mention, they share the same last name. And he starts asking questions.  Will he learn early on? Who will tell him? Would it change anything?
gpt
We know from Legends that Sidious was grooming the public perception of Anakin to be a post war hero. he was publicly seen as the only good Jedi, the one who did what had to be done, while not letting the Meek suffer needlessly. what's more after his assault on the Jedi Temple he effectively crushed a Jedi coup d'etat and saved not only the republic but the chancellor. Anakin losing large portions of his body mass, as well as his defining features was a terrible loss to the Sith cause. it's severely weakened anakin's connection to the force and effectively demolished all previous work done by Sidious towards the perception of Anakin. if you read the Darth plagueis novel it would be a similar master apprentice reflection of the book, we're Sidious was the background master puppeteer, and Anakin was the political Poster Child for the postwar era.

If Anakin had retained his Skywalker Monica, then it's entirely possible that through Imperial propaganda of anakin's face, and the extremely well known name of Skywalker, that the idea may have cropped up to young luke.