Wine is faster than Emscripten because asm.js is a slow target, not because HLE produces fundamentally better results than transpilation.
Wine has a variant, libwine, that can linked into a Windows binary at compile-time, replacing that Windows binary's linkages to Windows libraries with linkages to Wine. If you have the opportunity, this is always the more processor-efficient way to go (and I'm surprised so many companies choose to "port" their software to Linux using Cider rather than taking their source and compiling in libwine.)
If you didn't have the source of a Windows binary, you could create a Linux transpiler that replaces already-generated Windows symbols with Wine symbols statically while also potentially rewriting the ISA and calling conventions. The only reason one doesn't exist is that the overhead of running a very similar platform via HLE (i.e. one with the same ISA) is negligible. If OSX was still on PPC, for example, running Wine-on-PPC-OSX would work much better via transpilation than via HLE.
Speaking of PowerPC, a better candidate for "transpilation for performance purposes" would have been the PPC apps run under the Rosetta HLE emulator built into OSX. These apps could have been chewed-through once to produce a Carbon-linked x86 binary, that binary cached in the PPC binary's resource fork, and then executed immediately from then on. But Apple themselves had little reason to do this—they didn't want to give anyone an extra incentive to continue using PPC software instead of moving to x86/x64 software.