Amplifier current vrs watts; why is current more important?

Your typical amplifier outputs volts.

not quite right. Voltage (more correctly known as EMF, electro motive force) is the force that pushes energy out of the amp, down the cable, and into the speaker. Therefore, saying that an amp "outputs" volts is incorrect. I know it is confusing because you often see the term "output voltage", but that means the EMF measured at the output terminals, not that it is putting out volts.

Most amplifiers today are designed to act as much as possible like perfect voltage sources i.e. no matter how much energy you try to get out of the amp the voltage will remain constant. This is of course impossible, but what the original OP is referring to is how well a particular amp achieves this goal. There are also amplifiers designed to act as much as possible like perfect current sources including some of the offerings by Pass at First Watt

If an amp can maintain its voltage as the impedance drops then it has to deliver more current to do so. The better it is at doing this the "stiffer" it is. Conventional wisdom is that the stiffer an amp, all else being equal, the better it is. Of course, nothing is ever equal so whether or not this correlates to better performance is debatable and very dependent on the load the speaker presents to the amp.

Two power amplifiers that have the same power supply voltage may not have the same output current.

also not quite right. If 2 amps have identical output voltage into the same load they will have identical output current. It should say that 2 amps with the same power supply voltage may not be able to deliver the same amount of current. I know I am being picky but the lack of succinctness is what causes a lot of confusion among those with a limited knowledge of electronics. Amps do not output volts, they output current. Amps do not "have current", they output current.

Massive current gain stages are usually less linear than more modest stages and require more feedback to keep behaved

this is also incorrect. Typically an amp consists of a voltage amplifier or stages of voltage amplification followed by current gain from the final stage. This current stage is usually some variation on an emitter follower which is extremely linear. It is the previous voltage stages which tend to be non-linear where negative feedback is often employed to make them more linear. It is not unusual to have an amp with less unloaded voltage gain that is capable of delivering a lot more current into a low impedance load than one with a much higher unloaded voltage gain.

Surely you need to throw damping factor into the overall consideration too.

As someone who imports and sells a line focused on current vs. wattage, I have a POV. Art Audio amps are biased toward current. That is, our amps are high current and deliver lower wattage vs. many of our competitors. This means that with any given tube, we could deliver a much higher wattage number but opt against it.

This absolutely does allow for better handling of lower impedance speakers. But that is not the only reason and it is not just marketing.

The extra current delivers better frequency extremes. You get lighter & airier treble with better nuance and detail. Bass response is tighter and more refined. Maximize wattage in SET or Push Pull triode and treble gets rolled off and bass gets flabby.

You can find 35w SETs with 845 tubes. Why would Art Audio choose to produce an 18w amp when we could produce a 35w amp? The 35w amp will have flabby bass and rolled off treble while the 18w Carissa will deliver frequency extremes you woudn’t think are possible from a tube amp. But it certainly won’t drive every speaker.

That being said, the comment above where a kosst_amojan says ’it depends" really is true. It comes down to your speakers. I have seen a speaker with a high SPL of 98dB but has a minimum impedance of 3.2 ohms. Way harder to drive than you would think and even though 5w of power is enough to drive them in normal circumstances. You need 5 really good, current heavy watts.

Alternatively, I will be introducing a speaker (Verdant Blackthorn 1) with an 84dB SPL and a minimum impedance of 7.2 ohms. Not a difficult speaker to drive at all and current is of minimal concern. You just need enough power into 8 ohms to hit the volume you are looking for. 5w would deliver a whisper, 30w a bare minimum to have 10dB in headroom above 85dB.

I can say that my 18w Carissa could drive Focal Sopra Floor Standers mentioned by kosst_amojan with 15dB of headroom above 85dB. That same amp can’t drive my Blackthorn 1.

Alternatively, I have a 140w SS Class A/B amp that would do a wonderful job with my Blackthorn 1 but would blow a fuse if I tried to drive Sopras with it because it is barely stable at 4 ohms.