I see the relationship between BSPs and devices drivers as “has-a”. Board support packages include device drivers.
The differences between BSPs & kernels isn’t easy to distinguish. A kernel translates instructions to the hardware. Kernels are often written to particular families of hardware, so they’re not as portable or generic as they seem. It amounts to different permutations of the code for each architecture family.
The BSP acts as sort of the inverse: it provides the tools & instructions to work with that board’s specific set of hardware. In specific, controlled situations, the kernel could do this work. But the BSP enables any compatible kernel/OS/application stack to use that board, by following its configuration instructions.
If you just need to access CPU cycles & memory, maybe a few protocols (USB, Ethernet, and a couple of video types), a kernel with wide architecture support is fantastic, and there was a time when the breadth of that hardware abstraction was penultimately valued. But now, consider that the board may have a suite of sensors (accelerometer, magnetometer, gyroscope, light, proximity, atmospheric pressure, etc), telephony, there may be multiple CPUs, multiple GPUs, and so on. A kernel can be written to provide VGA, DVI, HDMI, DisplayPort, and several permutations of CPU/GPU combinations, if/when someone uses those particular hardware packages, but it’s not practical to write support for all theoretical contexts, compared to utilizing a BSP that’s built for a specific board. And even then, that would be for one kernel; the board is capable of supporting Linux, Windows, Android, Symbian, whatever.
That’s why efforts like Yocto exist, to further decouple kernel and hardware. BSPs make hardware sets extensible beyond a kernel, OS, and application stack or two, while kernels make a particular OS or application stack portable over multiple hardware architectures.