1.After delete one VM template in published content library, sync not work on subscribe content library by click synchronize action.
2.In order to update the status of VM templates in subscriber content library, publish action should be performed in publish content library refer to below doc:
https://techdocs.broadcom.com/us/en/vmware-cis/vsphere/vsphere/8-0/vsphere-virtual-machine-administration-guide-8-0/using-content-librariesvsphere-vm-admin/publish-the-contents-of-a-library-and-a-single-template-to-a-subscribervsphere-vm-admin.html
3.The update action spent much times on destroy all the existing VAPP template and clone the current ones to each subscriber content library.
VMware vCenter Server 8.x
By design
If need to decrease the waiting time on VAPP template update between publish content library and subscriber content library, DB level action could be another option, please contact to Broadcom support for more details.