It's an interesting question Don. The EL84 family of tubes have a huge following both in the production and reproduction crowds.
Personally, I think much (but not all) of the following is emotionally driven. That's hardly to say it's not real. But rather, to say that the following is largely driven by empirical evidence rather than scientific fact.
Certainly the production amplifier crowd has more scientific weight on their side, as in guitar amplifiers, different output tube families do produce their own unique clipping characteristics at the onset of overdrive (which can be displayed on a spectrum analyzer), so the love from that group for this tube is more scientifically based.
In reproduction amplifiers however, those designs intentionally and inherently are supposed to eliminate such colorations as different tubes or distortion characteristics might present, so that a 6L6 amplifier, a 6550 amplifier, and an EL84 amplifier should all in theory sound the same (all else being equal) with the exception of power output. And yet, we all have our preferences. Ultimately, at the bottom of it all, is the fact that our preferences are largely emotionally based, because let's face it, great amplifiers can be (and have been) built with all of these tubes. In the end, they can all produce very low distortion levels for the power they are capable of, and yet the preferences exist, which begs the question then that you pose: Is it the tubes themselves that sound different? Or, the amount of support circuitry (or lack of it) that each requires that (potentially) flavors the sound?
Certainly the bigger tubes (usually) take more support circuitry and larger power supplies, while the smaller tubes take less of everything. No doubt that is part of the perceived difference in sound that each tube produces.
I have designed many amplifiers using 2A3, 6V6, 6BQ5, 7591, 6L6, EL34, and 6550 tubes in SE, PP, triode, pentode, UL, and PPP configurations, operating in Class A, Class AB1, and Class AB2. By in large, within their design topology and power limits, they all sound about 95% the same -- which they darn well ought to if they are all achieving a high level of distortion, response, and output impedance performance. But that other all important 5% is what makes or breaks our preferences. Sometimes, it's supported with "facts" that can't be proven, or on whacked out theories that have (and could only) grown legs on the Internet, but ultimately, always on our own subjective -- and very real -- sonic impression that we all have. Just look how much Sony has been able to achieve in that last 5% with tube and component rolling.
I think in the end you nailed it with your big sound out of a small bottle description. The tubes are capable of very credible performance that can be scientifically demonstrated and is repeatable, but also have a likability factor that blows that remaining 5% out the top of the equation. That's about the best I can make of it, as I must admit, I am one of those who does appreciate a good EL84 based amplifier.
As for the X-202, it specifies 7189 tubes, but really, any EL84 family member will work just fine in these amplifiers. The 7189 specification was largely driven by Avery's want to be the first to use everything first, and that tube came out as this amplifier was on the drawing boards, so it got the nod.
People look at tube specifications as gospel, and to a large degree, they should. But they aren't absolute. A tube with a plate voltage rating of 300 volts certainly won't blow up with 310 volts applied to it, if the important ratings are adhered to, and otherwise good design practice is followed.
As a case in point, my SA-100 clone amplifier (with EFB) effectively applies 430 volts to the plates and 315 volts to the screens of the Gold Lion EL84 reissue tubes I installed in that amplifier when it was finished. Both of these voltage values exceed the 300 volt maximum limits published for these elements of these tubes. The tubes were new when installed, and have now just surpassed 1500 hours of use in that amplifier. Before they were installed, they were each tested for power output and recorded. Last month, when the 1500 hour mark was reached, the tubes were removed and tested again. All were still producing at least 96% of their new recorded power output level. Yet -- these tubes are operating beyond published voltage specs -- and yet, have clearly done just fine in spite of it. As a matter of fact, the paint and getter flash on these tubes has yet to show any real signs of aging at all!
Dave