/api/vAppTemplate/vappTemplate-853a4308-3ba1-46b8-bc6c-f6ffdf06d40c/ovf
This is exactly the same URL format you are using. There is no section for VMware tools (same or different then what you have).
did a get on /api/vAppTemplate/vappTemplate-853a4308-3ba1-46b8-bc6c-f6ffdf06d40c
But this is a group of VMs, and not an individual VM and doesn't show tools anything in it.
/api/vAppTemplate/vm-e2d53c8f-223e-4641-b49d-6d428b1f6c2b is a vm inside said vApp
this vm doesn't have tools installed, and I don't see a reference to tools.
/api/vAppTemplate/vm-1f92111b-1313-434a-a4ef-401bf4dae71e is a vm inside the same vApp template
this VM has tools installed as per the GUI, but no reference to tools here either.
I don't see a VirtualSystem reference in any of the XMLs I have received back.