I'll concede that "impossible" might be the wrong word choice, sorry. How about "impractical to the point of near impossibility"?
But I only have 22 years experience playing guitar and bass across genres as diverse as jazz, blues, gospel, metal, modern rock, and folk, so hey what do I know?
Well, allow me to go into a bit of science detail. The two polar opposites in amplification technology today are Class A and Class D amplifiers. Class A is the classic one, and any audiophile and sound engineer will swear by them on the quality metric. Class D is modern, based on transistors (usually MOSFETs), and while it dumps the linearity, it has a huge advantage in energy efficiency. Only in recent years has Class D become competitive in quality with the lower end of Class A technology, as the linearity of amplification makes a huge difference in high quality systems at higher amplification where the details pop out more.
Class A uses tubes, and works in the simplest and most accurate way by amplifying the complete signal. It maintains the most important characteristic needed to perfection - linearity of waveform amplification. When you try and get extra juice out of it, instead of the annoying clipping you get from digital amplification, you get a nicer sounding and more gradual reaction. The main drawback is energy efficiency. There's a theoretical peak efficiency of 50% to Class A amps, and what's worse is that it's usually less than half that. It makes tube amps that aren't small require massive heat sinks, and increases the BOM.
Class D is totally different. It dumps linearity out the window in favor of energy efficiency. It basically outputs a bunch of pulses, and then applies a low-pass filter to remove the annoying artifacts. While you might say that it's close enough - it isn't for high fidelity systems, especially in production studios, live performance, and audiophile cases. While competitive in quality with the low end of amplifiers (under $200), if you are looking for accurate detail reproduction there is no replacing linear amplification yet. The advantage is that even at the worst use scenarios, Class D amps have over 50% energy efficiency, and usually manage over 90% for higher volumes, making them need far less in the way of heat sinks which allows for leaner designs, and a better price/performance ratio.
The Class D amps will eventually win out. Class A amps have decades of R&D as an advantage, but Class D is improving by leaps and bounds. Eventually, Class D will become "good enough" and replace Class A altogether because of the advantages in price and energy efficiency. But that day has not yet come.
That being said, I use a class D amp for my home set up. It's close enough, and my power bill thanks me monthly.
Class A amplifiers can be made in solid state as well, very easily. As a truly simple, DIY example: http://headwize.com/?page_id=31
There's nothing about solid state that requires a circuit that "outputs a bunch of pulses, and then applies a low-pass filter" -- true linear amplification exists in the transistor domain just as well.
And even if solid state did require quantization (beyond the shot noise that vacuum tubes suffer from also), given that we can do direct digital synthesis up to the 100's of MHz regime pretty easily these days (i.e. beyond the bandwidth of most audio tube amplifiers), with 24+ bits of resolution, there's no reason that a solid DSP system couldn't emulate a vacuum tube response well enough to be electrically indistinguishable from the actual tube.
But, as the grandparent post notes, it's one heck of a lot easier/cheaper to just use vacuum tubes. Which is why I'm staring at a tube-based headphone amplifier on my desk right now. Though it does use solid state buffers, which incidentally are way more linear than the typical ways -- capacitors, transformers, etc. -- of removing the DC offset inherent to tubes...
There's a bit of misinformation in your post, but someone else already addressed it.
Personally I like Class D amps for basic uses like computer speakers and such. That said, I have a very basic class D receiver/amp that I use as a preamp for a larger 2.1 system with studio monitors for computer sound, as I like my FPS games and music to be clear and accurate while still being loud enough to "feel" things like explosions or timpanis.
Technically the Class D amp can push those monitors well enough, but at more than 1/3 volume there is some distortion and clipping, whereas on my discrete amp I have yet to find any distortion within comfortable listening levels.
True, but for some people "pretty good" isn't enough. As I said before, I've had some very nice solid state amps that I loved playing through. It's just that there are some sounds that have yet to be emulated properly via modern circuitry. Granted, it's all subjective; why does a Fender sound better or worse than a Gibson for a particular style of music? It all comes down to what kind of sound the musician wants to get out of his equipment, and if a solid state amp doesn't give him what he wants but a tube amp does, then I'd say the tube amp is still relevant.
Not impossible. Hard, expensive, or currently not available but not impossible.