For a while, I recommended 'em and used 'em myself. My own home PC was a Cyrix 6x86 P166+ for a year or two. Lovely machine -- a 133MHz processor that performed about 30-40% better than an Intel Pentium MMX at the same clock speed.
My then-employer, PC Pro magazine, recommended them too.
I only ever hit one problem: I had to turn down reviewing the latest version of Aldus PageMaker because it wouldn't run on a 6x86. I replaced it with a Baby-AT Slot A Gigabyte motherboard and a Pentium II 450. (Only the 100MHz front side bus Pentium IIs were worth bothering with IMHO. The 66MHz FSB PIIs could be outperformed by a cheaper SuperSocket 7 machine with a Cyrix chip.) It was very difficult to find a Baby-AT motherboard for a PII -- the market had switched to ATX by then -- but it allowed me to keep a case I particularly liked, and indeed, most of the components in that case, too.
The one single product that killed the Cyrix chips was id Software's Quake.
Quake used very cleverly optimised x86 code that interleaved FPU and integer instructions, as John Carmack had worked out that apart from instruction loading, which used the same registers, FPU and integer operations used different parts of the Pentium core and could effectively be overlapped. This nearly doubled the speed of FPU-intensive parts of the game's code.
The interleaving didn't work on Cyrix cores. It ran fine, but the operations did not overlap, so execution speed halved.
On every other benchmark and performance test we could devise, the 6x86 core was about 30-40% faster than the Intel Pentium core -- or the Pentium MMX, as nothing much used the extra instructions, so really only the additional L1 cache helped. (The Pentium 1 had 16 kB of L1; the Pentium MMX had 32 kB.)
But Quake was extremely popular, and everyone used it in their performance tests -- and thus hammered the Cyrix chips, even though the Cyrix was faster in ordinary use, in business/work/Windows operation, indeed in every other game except Quake.
And ultimately that killed Cyrix off. Shame, because the company had made some real improvements to the x86-32 design. Improving instructions-per-clock is more important than improving the raw clock speed, which was Intel's focus right up until the demise of the Netburst Pentium 4 line.
AMD with the 64-bit Sledgehammer core (Athlon 64 & Opteron) did the same to the P4 as Cyrix's 6x86 did to the Pentium 1. Indeed I have a vague memory some former Cyrix processor designers were involved.
Intel Israel came back with the (Pentium Pro-based) Pentium M line, intended for notebooks, and that led to the Core series, with IPC speeds that ultimately beat even AMD's. Today, nobody can touch Intel's high-end x86 CPUs. AMD is looking increasingly doomed, at least in that space. Sadly, though, Intel has surrendered the low end and is killing the Atom line.
The Atoms were always a bit gutless, but they were cheap, ran cool, and were frugal with power. In recent years they've enabled some interesting cheap low-end Windows 8 and Windows 10 tablets:
Given that there is Android for x86, and have already been Intel-powered Android phones, plus Windows 10 for phones today, this opened up the intriguing possibility of x86 Windows smartphones -- but then Intel slammed the door shut.
Cyrix still exists, but only as a brand for Via, with some very low-end x86 chips. Interestingly, these don't use Cyrix CPU cores -- they use a design taken from a different non-Intel x86 vendor, the IDT WinChip:
I installed a few WinChips as upgrades for low-speed Pentium PCs. The WinChip never was all that fast, but it was a very simple, stripped-down core, so it ran cool, was about as quick as a real Pentium core, but was cheaper and ran at higher clock speeds, so they were mainly sold as an aftermarket upgrade for tired old PCs. The Cyrix chips weren't a good fit for this, as they required different clock speeds, BIOS support, additional cooling and so on. IDT spotted a niche and exploited it, and oddly, that is the non-Intel x86 core that's survived at the low-end, and not the superior 6x86 one.
In the unlikely event that Via does some R&D work, it could potentially move into the space now vacated by the very low-power Atom chips. AMD is already strong in the low-end x86 desktop/notebook space with its Fusion processors which combine a 64-bit x86 core with an ATI-derived GPU, but they are too big, too hot-running and too power-hungry for smartphones or tablets.