To be fair, this sort of thing is going to become an elephant in the room eventually, and we might as well start discussing the matter. Granted, the OP handled the issue in a very tasteless manner, but the subject itself is a legitimate matter to ponder.
I feel the same. As Trek fans, we are all speculating about the issue. I don't think a policy of rigorous silence will make anything better. That said, the issue could have been introduced in a better way, the thread title for a start. I thought something like "The future without Anton" would be more sensitive.
Yelchin's death is a significant loss to Star Trek. He was not featured prominently in the movies, but he had a wide-eyed enthusiasm which was endearing and lifted the scenes he was in. Arguably, he was "Wesley done right". From that starting point, who knows where the character might have gone? Now we will never know.
All this said, Star Trek still exists and will continue into the future. What will happen in that future? I imagine the people behind the movies are starting to grapple with the same painful question.
Option 1: No more movies. I think only hardcore JJ haters would ultimately go with this. If Beyond makes a profit, a sequel is almost inevitable. It's possible some actors might resist the offer, but these are interesting parts that pay well, and I think they would probably conclude that continuing and doing their best would be a worthy tribute.
Option 2: Recast the part of Chekov. The two notable instances that come to my mind of recasting after an actor's death are Dumbledore in the Harry Potter movies, and Darren in the TV series Bewitched. Looking back, I don't think many regret that this was done. But that was before the rise to dominance of internet rage and the FEELINGS! of people on Twitter. I think Paramount probably won't recast the part for this iteration, due to fear of fan reaction.
Option 3: Chekov leaves the Enterprise, and is not replaced. I think this would leave a bit of a void in the cast, and focus too much attention on the efforts the writers have gone to to explain the change.
Option 4: Chekov leaves the Enterprise, and a new character is introduced. The pressure would be on to create an interesting new character with long-term potential. This could be done, but there would inevitably be some backlash based on dislike of the unfamiliar, and of the notion that the character or actor could ever be replaced.
I think option 4 is the most likely. It would reduce the sense of loss, and perhaps update the sex/racial balance of TOS. Best case scenario: A great new character would enter the Star Trek pantheon. I am not opposed to that. And if future artists return again to the adventures of Kirk and friends, I am sure there will always be a place for Chekov on board.