I would think your perception is incorrect, as the indexes do indeed change relative to the moves in specific stocks. You can set up a spreadsheet with the underlying stocks (in the correct percentages of course), use the index's divisor, and you will see that they are responsible for the indexes value.
I don't know what happens if a stock in the index is halted, as i know the index continues to change while a component is halted, and you can still trade options on that index. Whether they just inflate the price of the options OR approximate a value for that component of index i don't know.