Thank you very much for your quick and helpful reaction!
As I needed to indicate only the Labels as “clickable”, I adjusted the code as follows (for anyone who may be interested; is the method acceptable, by the way?):
...
if (Cesium.defined(pickedObject) && pickedObject.primitive) {
if (pickedObject.primitive instanceof Cesium.Label) {
element.style.cursor = 'pointer';
}
}
else {
...
Is it possible to suggest it as a new feature - to add “pointer” cursor option to API where relevant, as a rather standard hint for “clickability”?
Similarly confusing feature - well, for my “test users” 
- is seemingly inevitable popping-up of InfoBoxes from clicks on virtually all objects, including those not carrying any meaningful/author-added information.
Please, the same question again - is there any, preferably straight way for the Author/content creator to control which objects will have the “InfoBox feature” dis-/enabled?
Sorry I should probably ask the InfoBox question in another thread, but could not find proper one…
EDIT: The InfoBox problem is discussed here, but the presented answer does not seem to allow to disable InfoBox for individual object/s(?)