If the index instruments have a lot more liquidity than the underlying names, then that means they won't be able to gracefully absorb the liquidity shocks from an unwind in the index.
Here's some back of the envelope calculations. Between SPY, IVV, and VOO alone, there's $500 billion in S&P 500 index ETFs. Consider if an unwind event leads to 20% of index assets being redeemed in a single day. That's $100 billion from the above ETFs alone.
Now consider a typical thinly traded single-name stock like Chubb Limited (symbol CB). CB makes up 0.3% of the S&P 500 by weight. So in the hypothetical scenario above, the APs would have to collectively sell $300 million worth of CB in a single day. Chubb's entire ADV is only $238 million.
Trying to sell more than 100% of a stock's ADV in a single day is guaranteed to produce huge market impact. The current liquidity providers in CB almost certainly cannot absorb that amount of trading volume all in one direction. In that scenario, Chubb's stock might fall by 20%, for something that had nothing to do with the company itself.
I think the overall point is that a lot of single-name stocks nowadays don't really have much of an individual market. Names like TSLA, FB or TEVA definitely have a robust market with a lot of traders still focused on company specifics. But a lot of the more boring, lower volatility, mid-cap stocks (like CB) mostly just trade along with the index nowadays. If there are technicals related to index capital flows, stocks like that are going to get taken for a ride.
[1] https://www.etf.com/channels/sp-500-etfs [2] https://www.slickcharts.com/sp500 [3] https://finance.yahoo.com/quote/CB?p=CB