Why do certain security software require access to the kernel? To keep malware from getting to the kernel or something?
Security software doesn't necessarily NEED access to the kernel, but kernel-level access provides the maximum amount of access and visibility to the rest of the system. The only thing higher then kernel-level is hardware-level.
In the case of CrowdStrike, kernel-level access provides their software to have the highest privileges which yields in the most effective defense against malware (in theory). However third-party, kernel-level access is never a good idea. Software that has kernel-level access can be, and has been, exploited before. In the case of CrowdStrike, it was a faulty update that screwed over Windows systems. The more access you have in a system, the more you screw it over when something fails.
Doesn’t restricting access to the kernel offer more security?
Yes! You are correct. If implemented correctly of course, restricted access to the kernel provides a higher amount of security.
Wouldn’t malware also be unable to access the kernel?
In theory, the more restricted the kernel is, the more difficult it is for malware to access the kernel.
Kernel is what connects software and hardware, correct?
Yes. A function of the kernel is providing a way for software and hardware to communicate with each other.
I've come to this conclusion as well. I believe Apple has similar functionality with their "kernel-extensions", I believe it's called.