The old pose library should be removed in Blender 3.3, as discussed in the 2021-11-25 Animation & Rigging module meeting.

• Remove Python API marked as deprecated in T93405:
  • Operators:
    • poselib.action_sanitize
    • poselib.apply_pose
    • poselib.browse_interactive
    • poselib.new
    • poselib.pose_add
    • poselib.pose_move
    • poselib.pose_remove
    • poselib.pose_rename
    • poselib.unlink
  • RNA properties:
    • Object.pose_library
• Remove C/C++ code of the old pose library, except what's mentioned below. bAction::active_marker can be marked with DNA_DEPRECATED. it should stay because its future use is uncertain, and the impact of de-deprecating fields is unknown (at least to me, Sybren).
• Update the manual by removing the old pose library sections.

What should stay in Blender 3.3+:

• The DNA for action markers (bAction::markers) and the corresponding Python API. This will allow future versions of Blender to still convert old pose libraries to new ones.

T93405 tracks the Blender 3.2 targets: removal of the user interface and deprecation of the Python API.