You can play Nest of Thorns without matchmaking by opening Learn and going to Past Events / Earlier Events , then selecting the Crownfall archive entry there. That mode is accessible from the event archive, not only from queueing.

Quick steps

  1. Open Dota 2.
  2. Go to Learn.
  3. Open Past Events or Earlier Events.
  4. Select Crownfall.
  5. Launch Nest of Thorns from there.

Why this works

Players have reported that the mini-games are available in the Crownfall archive under Learn, so you do not need to sit in matchmaking just to start the mode. One discussion also notes the mini-games button should be available outside queue, but the archive path is the reliable workaround.

Small note

If you were trying to play it from the old queue-based mini-games flow, that setup appears to have changed or become inconsistent, which is why the archive route is the safer option.

TL;DR: go to Learn → Past Events / Earlier Events → Crownfall → Nest of Thorns.