I would like to be able to show some basic stats for my scene in BabylonJS, without needing to display the inspector. Is there a way to get those statistics from the scene?
Yes you can thanks to the scene instrumentation class:
var instrumentation = new BABYLON.SceneInstrumentation(scene);
Documentation: https://doc.babylonjs.com/divingDeeper/scene/optimize_your_scene#sceneinstrumentation
For mesh / faces / vertices count:
scene.getTotalVertices(); // total vertices
scene.getActiveIndices() / 3; // faces
scene.getActiveMeshes().length; // Total meshes