Unfortunately, Google Drive and Dropbox only partially participate in Android's Storage Access Framework. In Aegis, exporting only requires the creation of a file, so that works with both. Configuring backups on the other hand requires selecting a folder, but most cloud providers don't support that. A notable exception is Nextcloud.
How about cloud paas providers like AWS? The user could generate a IAM access key with permissions to manage a specific bucket and configure Aegis with the key. Aegis would use the cloud Api to upload the backup.
If that is of interest, I could help implement that.
commented on my question with a snipped I wrote based on that extraction method from authy. The code generates a Aegis compatible database instead of printing QR codes