Depends on how far the Blue Jays go. If they make it to the ALCS lets say, that would be a huge positive with this young group. I'm sure Rogers would view the Blue Jays as really close to winning a World Series in 2023 and 2024, so doubt Atkins would go anywhere.
The 2016 Blue Jays were a carry over from the 2015 Blue Jays, so the rebuild didn't start until really things blew up in 2017. Atkins/Shapiro took about 3 seasons to get this team to where it needed to be in terms of contenders. 2020 season was a first taste, though doesn't count IMO. Blue Jays in 2021 were serious contenders who missed out by ONE game. Sucks because a lot of games they blew in April/May/June could have been the difference with a stronger pen. The window to contend is over the next 2-3 years. If the Blue Jays don't win a World Series by 2025 with this core in place, the window will close on them most likely.
If the Blue Jays fail to reach the Postseason and/or lose in the Wildcard Round, and the glaring issue was because of the bullpen blowing games, then I think Atkins can be on the hot seat. Though still think he would get a pass for 2023. If the Jays go into 2023 with an awful bullpen again and it costs them winning the division especially with this young core they have, then yes the front office should be held accountable.