Why "this phrase is unlikely to have ever been part of the ideology of the mysterious sect or sects that owned the scrolls"?

In short, the quote first appears in Sifrei Devarim, which is a tannaitic midrash from the school of Rabbi Akiva. This means the materials that compose it were authored not long after the destruction of the temple. In other words, there's a pretty good chance that these sages were familiar with the DSS sect(s). While there are arguments that these groups' ideology did not contradict that of the sages and may have even been part of the same circle,

there are many arguments to support that these were fringe groups and indeed quite a few of their materials differ greatly from what we know of the sages. It's speculative on my part, but I feel that it's unlikely that the sages would have quoted something from a fringe sect in such a positive matter.

If you're interested in an opinion that these authors were mainstream, you can see for example Neil Asher Silberman's The Hidden Scrolls (although his view is interspersed with explaining the politics surrounding the research of the scrolls).

AFAIK, DSS comprise a library of scrolls from very different schools. They saw themselves as "keepers" rather than "teachers" - e.g. they preserved different versions of the same books. If the "Scroll of Hassidim" existed, they would probably lay their hands on it. Regarding ideology, how do you know if a "scriptural" verse is/aligns with ideology?

Regarding Hassidim Rishonim, I'm not sure if they were a sect or an organized movement at all, I see them more as single individuals remembered by the Pharisees as extra-pious.

@AlBerko your views on the role of the DSS people are also debatable. I stated my own opinion on the matter. I don't mind debating the issue but the first paragraph of my answer is the main point and that answered my own question.

@Harel13 I only meant to point out that the DSS is a library, so if a scroll was popular enough we could expect it to be found in Qumran. Who are "they" that you equate to אנשי כנסת הגדלה?

IIRC, Shimon [allegedly] greeted Alexander the Great, centuries before Jewish sectarianism emerged.

They I meant Chassidim Rishonim. It's the gist of what I wrote in that old paper.

I don't mean they were sectarians, but quite the opposite.

@Harel13 Any sources for the existence of the כנסת הגדולה?

@AlBerko several Chazalic sources which I'm sure you're familiar with. Some of those sources as well as later ones identify them with figures from the books of Ezra, Nechemiah, Chaggai, Zechariah and Malachi. What else do you have in mind?

@Harel13 Right, I thought you had something substantial. כנסת הגדלה is more of a mythical institution, just like the Great Sanhedrin. The very name כנסת הגדלה points to its Mishnaic origin. It is possible that Jews had some governmental independence and therefore some institutions, but those were nothing like "pre-sanhedrin".

@AlBerko your claims are equally or even less substantiable. I'm familiar enough with where your views have gone in recent years. I, however, disagree and tend to put more trust in our sources.

OK, then. So how do you see the Qumranites?

@AlBerko I don't have a very well-developed opinion on them, but as I said, right now I'm leaning to them being sectarians, although I'm open to arguments in favor of their being mainstream.

OK, nice talking to you! If you have more interesting theories let me know.

@AlBerko I actually have a lot, but sadly I don't have time to write down all of them. Some have to do with Jewish and Jewish-Christian sects too.