Why doesn't Linux use the hardware context switch via the TSS?

The x86 TSS is very slow for hardware multitasking and offers almost no benefits when compared to software task switching. (In fact, I think doing it manually beats the TSS a lot of times)

The TSS is known also for being annoying and tedious to work with and it is not portable, even to x86-64. Linux aims at working on multiple architectures so they probably opted to use software task switching because it can be written in a machine independent way. Also, Software task switching provides a lot more power over what can be done and is generally easier to setup than the TSS is.

I believe Windows 3.1 used the TSS, but at least the NT >5 kernel does not. I do not know of any Unix-like OS that uses the TSS.

Do note that the TSS is mandatory. The thing that OSs do though is create a single TSS entry(per processor) and everytime they need to switch tasks, they just change out this single TSS. And also the only fields used in the TSS by software task switching is ESP0 and SS0. This is used to get to ring 0 from ring 3 code for interrupts. Without a TSS, there would be no known Ring 0 stack which would of course lead to a GPF and eventually triple fault.

Linux used to use HW-based switching, in the pre-1.3 timeframe iirc. I believe sw-based context switching turned out to be faster, and it is more flexible.

Another reason may have been minimizing arch-specific code. The first port of Linux to a non-x86 architecture was Alpha. Alpha didn't have TSS, so more code could be shared if all archs used SW switching. (Just a guess.) Unfortunately the kernel changelogs for the 1.2-1.3 kernel period are not well-preserved, so I can't be more specific.

Linux doesn't use a segmented memory model, so this segmentation specific feature isn't used.

x86 CPUs have many different kinds of hardware support for context switching, so the distinction isn't hardware vs software, but more how does an OS use the various hardware features available. It isn't necessary to use them all.

Linux is so efficiency focussed that you can bet that someone has profiled every option that is possible, and that the options currently used are the best available compromise.