The most famous case I know of that stems from a publicly available SAR is former Speaker of the House Denny Hastert (R-Il) after his retirement from the House, when he was a lobbyist. He was being blackmailed by someone for his previous sexual assaults from his days as a high school teacher and wrestling coach three decades earlier, before he went into politics. He went to a bank to withdraw a bunch of cash to pay his blackmailer, and the bank started to fill out the form they do whenever you withdraw over $10,000 in cash (not a SAR, just a regular form). He saw the form being filled out, and decided he didn't want a paper trail with the government showing he was being blackmailed. So instead he stood in line hundreds of times at the bank, withdrawing $5,000 each time. This was why a SAR was filed, because he was clearly doing something called "structuring" which is setting up transactions deliberately to avoid those disclosure rules. After the FBI investigated, they decided that they couldn't get him for child sexual abuse, because it had been so long ago that the statue of limitations had expired. But they could get him for structuring, to which he plead guilty and served 15 months in jail.

Besides the fact that knowing about when they are filed would tip off criminals, there is also the fact that oftentimes they are filed for perfectly innocuous reasons, and are never investigated and don't go anywhere. Not ideal to have publicly available "this guy did something suspicious" flags that have not been investigated further looming over you and haunting the rest of your life. Since SAR's are not shared with other financial institutions, they won't follow you the same way.