Why do some amplifiers throw a bigger soundstage than others?

It doesn't happen. It's a placebo. The only reason you'd hear a difference is if one amp is underpowered and the other has more power to drive the speaker, assuming the speaker can handle it.

Should spatial perception x what you describe differ for, say, bass in outdoor setups (I am sorry - I realize this is slightly tangential from the OP query)?

@benanders Outdoors we can sort out where bass notes come from. But in most rooms indoors we cannot since it takes a few iterations of the bass note to pass our ears before we can know what bass note it is. By that time (in most rooms) the bass has bounced off of several walls and so is 100% reverberant.

Stereophile Review, Pass Labs XA60.8, Jim Austin, November 22, 2017

Nelson Pass said in my interview with him in the September 2017 issue. He noted how his First Watt experiments with amplifiers based on static-induction transistors (SITs) led to an insight into the subjective effects of second-harmonic distortion, particularly its phase, and in turn influenced his design of his big Pass Labs amps. "The SIT being very much like a triode, it is easy to make a single bias adjustment which affects the second harmonic distortion of the device, ranging from a relatively large amount [of] positive phase second [harmonic] through a null point with no second [harmonic], to large, negative phase second-harmonic distortion," he said. "Negative-phase second harmonic tends to expand the perception of front-to-back space in the soundstage, separating instruments a bit. Positive phase does the opposite, putting things subjectively closer and 'in your face.'"

Very interesting thanks 

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+1 noromance

it’s on the record.  On certain LPs I can hear people talking to my right of my seating position. There is a party on a Donald Fagan album. People are clapping and talking around me  

Bent