Is it not the other way around, that KVM runs on bare metal (and needs processor support) while Xen runs as processes (and needs special kernel binaries)?
It's true that KVM needs processor support: it kind of adds a special process type that the kernel runs in an virtualized environment, through the hardware virtualization features. The linux kernel of the host schedules the execution of the VMs.
Xen has a small hypervisor running on bare metal. It can run both unmodified guests using hardware support or modified guests where hardware access is replaced with direct calls into the hypervisor (paravirtualization). The small hypervisor schedules the execution of the VMs. For access to devices it cooperates with a special virtual machine (dom0), which has full access to the hardware, runs the drivers and multiplexes access for the other VMs - the hypervisor is really primarily scheduling and passing data between domains, very micro-kernel like. Dom0 needs kernel features to fulfill that role.
Though, really, I'm just quoting Anthony Liguori from 6 years ago, so credit where it's due: