Because Loki became a variant as a direct consequence of the Avengers' (albeit bungled) meddling in one timeline, it stands to reason those timelines would end up pruned, as they diverge from the "sacred" timeline (an escaped Loki and a spoiler'd and humiliated cap in one, a time-removed Thanos in another, etc.). If time traveling to undo Thanos' quasi-omnicide was part of the 'sacred' timeline, were the other timelines set to be pruned by someone's intent?
No. What the Avengers did was supposed to happen, according to He Who Remains, via the TVA / Judge Ravonna Renslayer. Judge Renslayer confirmed that the Avengers' actions were part of the TVA's plan. The only reason Loki became a variant was because he escaped with the Tesseract, which was not supposed to happen*. The alternate 2012 timeline where Loki escaped was already pruned by the TVA, as shown in Loki S01E01.
* This is according to the TVA / Judge Renslayer
but according to Kang / He Who Remains, he planned for this to happen — to have Loki and Sylvie lead the TVA.
Relevant dialogue from Loki S01E01 "Glorious Purpose":
Judge Renslayer (talking to Loki): We're not here to talk about the Avengers.
Judge Renslayer: What they did was supposed to happen. You escaping was not.
One way to understand the Sacred Timeline is not as a single line, but as a bundle of strands that all follow a similar direction. These strands represent different timelines that are allowed to branch off slightly from each other as long as they don’t deviate too much from the main flow, and cause a multiversal war via a nexus event. You can read more about this on Science Fiction & Fantasy SE: “If there’s only one Sacred Timeline, then why do multiple variants of Loki exist?”
Being a branched timeline doesn't necessarily mean that it will be pruned by the TVA. The TVA only prunes a timeline if a nexus event occurs in it. Because if that timeline is left unpruned, it will lead to another multiversal war. Otherwise, the TVA leaves the timeline unpruned.