I agree with your conclusion, Scooter.
However, there can certainly be ambiance. Just like any recording incorporating microphones rather or in-addition to direct-in, the amount of ambiance in a stack-tape is going to vary depending on how loud the PA is at that position in comparison to the loudness of the room ambiance and audience reaction, primarily. Yes the PA is going to be significantly louder at a close position in comparison to a farther one, which if significant enough can mask audibility of the ambiance/audience content when the band is playing, or it may just make that content less audible. Even if no ambiance/audience is audible at all while the band is playing, it will be audible when the band is not playing, although at a lower level than as would be heard from farther away (by whatever amount you need to set recording gain lower to keep the PA content from clipping). It's essentially the same as the experience of standing there listening from that position.
The two keys to making a good stack tape are the same as making a good AUD: position and position. In this case position in terms of getting a good frequency balance from the PA (as discussed above) and position in terms of getting the desired ratio of PA sound verses room/audience sound.
Also, the recording will not have the same stereo qualities as an AUD, stage-tape or non-mono SBD, but will not be mono. The ambient/audience portion will be stereo, and depending on how audible that portion is, will render the close-PA sound in stereo. That gets to the root of the similar question that sometimes pops up around TS- "why use two mics to record any acoustic solo performer?" Even though the direct sound arriving from the performer may not have stereo qualities, performances are never held in anechoic rooms.
I also find stereo-mic stack-taped PA sound not to be truly mono on its own even when it is effectively acoustically masking all ambiant/audience sound. There is always some response variation between the microphone pair due to configuration, pattern, slight movement if worn by a human mic-stand, etc. This is the case even very close to the PA, in which the difference can be related to the frequency-blending issue discussed earlier in the thread. Take any stereo recorded stack-tape and listen to the difference when switching between stereo and mono playback. There is always a difference in my esperience.
Finally, this method works in small clubs as well as large arenas. In smaller clubs not everything may be represented in the PA, so take care to get the balance of PA sound, stage sound, and room sound right at the stack-recording position. I tend to think of stack-taping not so much as an isolated specific method in itself, rather than one extreme of a continuum of recording positions with a few different dimensions to it: Centered in the traditional sweet-spot over to close-stack being this particular dimension. Distant room recording location up to on-stage being another. Choosing to setup off-center to get on-axis and perhaps closer to PA so as to achieve sufficient clarity while potentially sacrificing some direct PA stereo image quality is an example of moving along this continuum and a bit part of the art of taping.