Qubes doesn't use a storage domain, so the SATA controller (or in your case, NVME) doesn't use pciback as far as I know. If not for that, I would say it's probably because pciback blocks writes to read-only config space of PCI devices. However storage devices in Qubes just use the normal driver in dom0, just like any other Linux. You can check with `lspci -k`. Unfortunately I don't know enough about NVME to be much help.
Have you tried installing Xen in any other distro? See if the problem occurs there. You can also test suspend/resume while you're at it. If you find that it fixes the NVME problem and suspend works, you can continue using that distro under Xen, even if you don't actually use any domUs.
No, pciback is like a proxy server that communicates via Xen mechanisms to domU's that want to use the hardware. Dom0 itself talks directly to the hardware just like a regular OS (for the most part, though there are exceptions).
"Domain 0 has responsibility for all devices on the system. Normally, as it discovers PCI devices, it passes those to drivers within the Linux kernel. In order for a device to be accessed by a guest, the device must instead be assigned to a special domain 0 driver. This driver is called xen-pciback in pvops kernels, and called pciback in classic kernels. PV guests access the device via a kernel driver in the guest called xen-pcifront (pcifront in classic xen kernels), which connects to pciback. HVM guests see the device on the emulated PCI bus presented by QEMU. "
https://wiki.xenproject.org/wiki/Xen_PCI_Passthrough
As you can see, your NVMe controller is using the "nvme" driver, not pciback. Xen actually does support passthru of SATA/NVMe/SCSI/whatever controllers to domUs, referred to as a storage domain, but Qubes doesn't utilize this feature.
Anyway, getting back to the point, as I said there are certain exceptions where Xen intervenes between dom0 and hardware sometimes. An example would be the IOMMU, which Xen takes over early in the boot process, and hides much of its functionality from the dom0 OS. I don't know if there might be exceptions like this in the context of PCI devices or NVMe controllers in particular. You'd have to ask someone more familiar with the inner workings of Xen. Usually Xen breaks things rather than fixing them, but I suppose it's possible that Xen is protecting your NVMe controller from the OS in some way, whether intentionally or not.