I don't think that's a very good mental model for kernel vs user. I think it's useful to look at the implementation of these abstractions in order to fully understand them:
What is a Kernel?
A kernel is basically just a piece of memory. It was privileged enough to be loaded before anything else, thereby allowing it to set the CPU's interrupt vectors.
Interrupts control everything, including I/O, timers, and virtual memory. That means that the kernel gets to decide how all that is handled.
A library is also just a piece of memory, and you can very well look at the kernel as the "system call library", among other things. But because the kernel represents the hardware, that piece of memory is shared among everyone.
Kernel Mode vs User Mode
Kernel mode is the CPU's "natural" mode, with no restrictions (on x86 CPUS - "ring 0"). User mode (on x86 CPUs - "ring 3") is when the CPU is instructed to trigger an interrupt whenever certain instructions are used or whenever some memory locations are accessed. This allows the kernel to have the CPU execute specific kernel code when the user tries to access kernel memory or memory representing I/O ports or hardware memory such as the GPU's frame buffer.
Processes and Threads
A process is also just a piece of memory, consisting of its own heap and the memory used by libraries, among which is the kernel.
A thread (= a unit of scheduling) is just a stack with an ID that the kernel knows of and tracks. That's the call stack that the CPU uses when the thread is running. User threads have 2 stacks: one for user mode and one for kernel mode - but they still have the same ID.
Because the kernel controls timers, it sets up a timer to go off e.g. every 1 ms. When the timer triggers ("timer interrupt"), the CPU runs the callback that the kernel set up for that interrupt, where the kernel can see that the current thread has been running for a while and decide to unschedule it and schedule another thread instead.
Virtual Memory Context
By "virtual memory context" I mean all the memory that can be accessed by the CPU. This includes all the memory of the process - including the user-mode heap and memory of libraries, user-mode call stacks of all process threads, kernel-mode stack of all threads in the system, the kernel's heap memory, I/O ports, and hardware memory.
When an interrupt or a system call occur, the virtual memory context doesn't change, only a CPU flag is flipped (i.e. from ring 3 to ring 0) and the CPU is now back in its "natural" kernel mode where it can freely access kernel memory, I/O ports and hardware memory.
When a new process is created, what actually happens is that a new thread is created, and assigned a new virtual memory context. Therefore, every process starts as single-threaded. That thread can later ask the kernel via a system call to create more threads (= stacks) which share its virtual memory context (= process), or ask the kernel to create more threads, each with a new virtual memory context (= new processes).
Kernel Threads
Like any other library, the kernel can have its own background threads for optimization purposes. When such a need arises (which can happen in the memory context of any process when servicing a system call), the kernel will create new threads and give them a special memory context, which is a context that only contains the kernel's memory, with no access to memory of any process.