"The questions all are phrased so no system in particular is mentioned"
The problem, here, is that sometimes those questions are phrased in such a way that people assume they're referring to a particular system. This is rare so far - for the most part, people are seem more likely to ask questions which could be answered in a system-agnostic way, but are narrowing the scope to a particular system and getting more focused answers as a result. Still, if someone really wanted to ask Is there any benefit to trying to have someone start at the bottom and joining in? in a system-agnostic context, they'd pretty much have to disclaim that they were talking about D&D at some point, either in the question, or in response to a comment asking, "Are you talking about D&D?" The system-agnostic tag strikes me as an excellent way to formally include that disclaimer: "I really do want something that applies to all systems [which have mechanics similar to the one I'm discussing]."
Indeed, [dnd] is seeing a LOT of use as a shorthand for "systems [which have D&D-like mechanics]." With that in mind, I could see tagging a question "[dnd] [system-agnostic]" to indicate that you're talking about games that act like D&D, but that you don't believe the answer depends on any particular system.
"When you're searching for an answer (especially those that are new to the site), your first thought for searching is generally going to be specific to the system you're in."
Well, we should plan tags in such a way that they're useful to people who are not new to the site, as well. And if you have a question that you know will be the same no matter what the system, you might well include [system-agnostic] to avoid getting answers that are too cluttered with a particular system's mechanics. More than that, though, I think that [system-agnostic] is a fascinating tag to browse, because it has answers that are by design applicable regardless of what system I might choose. That strikes me as well-defined and useful.
"The only value it may have is for those that are just browsing the site looking for nothing in particular, just seeing what's all out there."
I do not agree that this activity is worthless. The top-voted [system-agnostic] questions are a fascinating collection already, and I look forward to continuing to browse them as the site grows.
Another problem I've been made aware of:
"There is disagreement about what [system-agnostic] actually means."
Jeremiah Genest believes it means something very specific, but hasn't shared that definition. I took it to mean something very different. So I'll admit that [system-agnostic] may not be the best possible wording to achieve what I want, because it generates such ideological conflict. Here's what I want:
I want a tag that includes "What problems do evil characters cause?" "What iPhone apps do you use for gaming?" and "What are some alternatives to miniatures?" but does not include "Do healing bonuses (Pacifist healer) apply to flat healing effects like consecrated ground?" and "What are the exact rolls and totals for a direct combat spell?" Each of the first three questions clearly cannot possibly apply to ALL game systems imaginable. I propose that no question could do so. I don't want a tag that means that. I want a tag that means "my question has broad applicability across a number of systems." I don't care what it's called, but that seems to be what people are using [system-agnostic] to mean, and I appreciate being able to readily find the questions that (some people feel) are "system-agnostic" (as opposed to truly being "system agnostic").