Updated to Monterey 12.4 and external hard drive can't be opened in AE

You've probably tried this, but have you checked the permissions on the drive?  I've had a few of my HDs default to read only after OS updates....

I am not sure if what’s described below is what’s causing this specific problem, but it is a common reason why Mac applications are blocked from write access.

If the “update” you applied to the Mac was a upgrade to one of the most recent major versions such as macOS 12 Monterey, go to the Security & Privacy system preferences panel, Privacy tab, and check the settings for Full Disk Access and Files and Folders.

To defend more tightly against malware, applications start out with restricted access, and you can grant permission outward from there as needed. When permission is not granted to an application, it may be blocked from accessing some folders or volumes. 

If you trust Adobe, grant Full Disk Access permission to After Effects so that macOS lets it save anywhere. If you want to grant permission to just some folders or volumes, grant it when macOS asks you, and that scope will be added to the Files and Folders permissions.

Did this drive work OK with a Mac before? If it was formatted for Windows you may not be able to save to the drive.