I believe there are loopers using Live and other tools onstage, but one of the great beauties of the looping community is the breadth of approaches that are seen. Thus, the question is not as simple as whether or not a looper chooses to use a tempo. It could be that the effects they're going for are not quite as obvious as you think.
First of all, many loopers do not choose to work in tempo, they are more concerned with textures than rhythms.
Secondly, even those loopers that do work in tempo may not wish to predetermine the tempo (even as far as setting it immediately before starting a piece), but to leave it as an improvisational attribute. Rick Walker, for example, may do a piece based on orange plastic items or whatever, and those objects may constitute the sole consistent aspect of his "orange plastic" piece. Or he may even have motifs he uses consistently for a piece, but tempo and interpretation of the motifs might be things that are determined by his feeling on that night. He may wish to layer loops that do not have a straightforward tempo relationship or, indeed, any relationship at all having to do with tempo or meter.
There are also loopers who feel that dedicated hardware devices are a more fluid looping environment than dealing with a laptop, where one has to look at a complex screen and set up external hardware controllers with the software, as opposed to those functions being inherent in the device. If you're looking at display, you can't be looking at an instrument you're playing or anything else. It also dictates your physical position onstage: you have to be able to see the computer, whether or not you need to touch it to operate it. Sometimes it's just more *fun* to play an instrument that does not have the "rules" of a personal computer.
Further, there are many that dislike using a general-purpose OS in such a specialized live performance environment, or simply don't trust Windows (being the dominant OS) in an onstage context. Yes, I know lots of people use laptops in live performance every day, but that doesn't mean problems never occur. (Of course, problems can occur with dedicated hardware, too, but it won't be because some software entirely unrelated to a music application somehow got fouled.)
Having said all of that, there are loopers that use MIDI clock, especially to synchronize with another looper's rig, yet that still does not necessarily mean a time signature is used.
In short, while tools like Ableton Live are very powerful, they may function based on a paradigm, such as the predetermination of tempo and meter, that imposes undesirable constraints on a given looper's music.