Time Machine backup on new Mac not overwriting old Mac backups

You cannot direct TM to overwrite old backups. And as you posted, TM does not always seem to do the "deleting old backups" thing very consistently. You could have allowed the new Mac to inherit the old backups when you first set it up, but since you did not at that time, the option is no longer on the table.

If your new Mac inherits your backup history - Apple Support

You can, however, simply erase the backup drive and allow TM to create a new backup set on the drive.

Before you erase the drive for this, you would be wise to create a backup on another drive to have on hand should something go sideways in the meantime before TM completes the first new backup.

+1

If the external drive is less than 2 years old, then you have the option to just continue on with the inherited backups or erase the external drive and start new Time Machine backups.

However if the external drive is more than 2 or 3 years old, then you should consider replacing the old external drive with a new external drive for your Time Machine backups.