Class A - don't let them pull the wool over your eyes
The first thing we need to realize is that you can't just talk about Class A without talking about single-ended and complementary circuits too.
A single-ended circuit is one in which the power rail is some (typically) positive voltage and the the other side of the power supply is tied to earth and is therefore zero volts.
Let's say that the positive power rail is at 20 volts. This means that the output has to be biased to 10 volts when there is no signal. This is so that the signal can swing positively by almost 10 volts, and swing in the opposite direction by almost 10 volts too.
Since the output has a DC component of 10 volts, this must be isolated by a capacitor or transformer.
The problem with single-ended design is that the amplifying device, let's assume a transistor, is not symmetrical in its characteristics. It will perform differently when the signal is close to zero volts than when it is close to 20 volts.
There will be distortion, and the distortion will be different on the positive-going and negative-going half-cycles of the waveform.
To supply the current for the output of a single-ended amplifier, there must always be current flowing through the output device - enough current to supply whatever the amplifier is connected to, whatever the output voltage. When a circuit has such a 'standing current', it is described as being Class A.
Class A does not mean single-ended, it means having a standing current.
On the other hand a circuit can be designed which is complementary. It has positive and negative power rails, typically +20 volts and -20 volts. Earth, as before, is naturally at zero volts.
A complementary amplifier output has one transistor for the positive-going half-cycles of the waveform, another transistor for the negative-going half-cycles.
Since this circuit is symmetrical about zero volts, the distortions produced are symmetrical too. This isn't to say that they cancel out, but at least the positive- and negative-going half-cycles distort in the same way.
But where a single-ended circuit must be Class A - there is no other option - a complementary circuit can be Class B. This means that on the positive-going half-cycles, the transistor that handles negative-going half-cycles is switched off, allowing no current to flow, and vice-versa.
In practice there has to be a small standing current, otherwise there will be distortion at the crossover point. So a complementary output stage can be so-called Class AB, meaning that it inclines towards Class B, where the transistors are alternately switched off. But there is an element of Class A in that there is a small standing current, but not nearly enough to completely supply the output.
But a complementary circuit can be Class A if you want. The standing current is increased so that neither transistor ever switches off.
A complementary can be connected directly to the load without a capacitor or inductor, since there is no DC component in the output.
For a power amplifier, it is really only practical to use Class AB. To use Class A means having a standing current of several amperes, which is grossly inefficient. There are some esoteric hi-fi Class A amplifiers, but they are either of low power, or they are huge (and hot!).
For other amplification stages used in audio, and there are many, it is most practical to use complementary outputs. Since the current requirements are not high, they can easily be Class A. There is no reason to do otherwise (although having no reason doesn't mean it isn't done!).
Class A complementary amplifiers can be of extremely low distortion and very clean. However, it is often thought that a single-ended amplifier (which of course must be Class A), produces distortion that is subjectively pleasing. Hence their continuing popularity.
And if single-ended Class A was good enough for Rupert Neve, then it certainly must have something going for it.
But really, it isn't rocket science.