Immutable Backups how are you currently restoring from S3? Do you copy the files from S3 down to local disk and then perform a restore, or do you have some way to restore directly from S3?
Using the new backup format we store things in archive files, so there won't be a list of files that match what you have locally in the cloud. The way you have things set up, you may want to have different backup plans for each database, but I think you could include the t-log backups if you wanted since most backups are removed every week.
As a side-note, if you're deleting your backups locally after the next successful full backup, then you're only leaving yourself with one days' worth of backups since you're deleting the previous week. But if that process is working for you, that's fine. I would be a little concerned about that, especially since you're backing up locally and presumably you have disk space. Restoring from the cloud is certainly going to take a lot longer, and if you find yourselves restoring a few times a week, I'd be looking to keep more backups locally to avoid the need to restore from the cloud. But more backups kept locally means larger full backups on our side, so there's a trade-off.
You can run the fulls as frequently or as infrequently as you want. I assume you don't need all backups for the last 10 years, is that correct? Or are you keeping every single full and every single differential for every single database for 10 years? I guess I would need to know that to provide some guidance on retention settings.
If you're truly keeping 10 years worth of everything, then you don't need to run the fulls very frequently at all. And you may want to run them after the cleanup happens locally to minimize the next full backup size on the S3. You could run them once a month if you wanted. Or you could run them weekly if they're aligned with the local full backup and cleanup. In that case you just going to have a new full backup for the database and the rest of the folder it sounds like it's going to be cleared out anyway. So the synthetic won't really do anything as the only thing being backed up is the new full backup because that's the only file in the folder. And in that sense, a full is no different in what data is backed up than an incremental.
Use the Keep Backups For as an absolute measure of how long you need to keep the backups. Use the GFS settings if you do not need to keep every backup and are looking for longer-term retention, but less granular in restore points - or use GFS if you need immutability.