The problem with continuing the chronological nature of the character to account for Harrison's age, is that it has to evolve into some other genre in order to mesh with the era in which the story takes place and that's a problem because Indy's time period is firmly planted in the golden era of film. That's why Crystal Skull felt off. It took place outside that context. You can't just uproot the concept and modernize it in a different era when it's supposed to take place in the real world. Star Wars could do it because the setting of the story was completely fictionalized. Indy doesn't have that luxury. Those types of movies weren't filled with subversive ideas. They were in made in total opposition to that. They were made to escape from it.
As for recasting the role and setting it once again back in the 1930's, modern audiences are too cynical to really identify with the archetypal adventure stories set in the relatively unknown world. If you need further proof of this, just look at the countless articles by entertainment journalists over the last decade that have a habit of deconstructing classic movies to the point of outright mockery, all the while claiming to be huge fans. The irony is staggering. Is everyone a hipster? Treasure hunting stories often have elements that most would find uncomfortable and if the story was going to stay true to those roots it's only natural to pay homage to those tropes. Though now those things would be considered problematic and it's a risk to not address certain cultural insensitivities because, you know, reasons. Modern audiences don't seem able to put things in context. Hollywood isn't exactly known for handling such things with any tact either, so I'm not at all interested in seeing more of that. Some aspects of movies don't age well but that's evidence that movies are a product of their time and if you're going to enjoy older movies you have to have at least a rudimentary understanding of this.
Plus how much can you really mine from Indiana Jones? I mean really? The concept worked great for 3 films and even Spielberg only did Crystal Skull to appease George. You can only have Indy fighting Nazi's so many times that it gets tired. It worked in Last Crusade, but barely. I love that movie but it was kind of a retread of Raiders without the same punch. Do fans really want to see a further watered down Indiana Jones? I don't.