Thomas Cook is a fine example of the agency effect

Each entity that you have involved in your transaction is an opportunity for something to go wrong. Perhaps you see a price listed on CheapOAir, Kayak, Orbitz or whatever. You are trusting that they will properly remit payment on your behalf. Even if they do, if you have a problem with the service provider (airline, hotel, etc.), the agency is the customer, not you. It's best to get the price from the agency and contact the service provider directly with the price you have in mind. They usually can come pretty close. Don't mention the agency name specifically or the service provider could contractually be prevented from offering you a deal. Just say that you found the price "online".

If you’re talking about 3rd party booking sites as a general rule then yes, this is a strategum that can work to get you a deal.

If for example you find a deal on Expedia for a hotel that’s $99, the hotel depending on their contract can be losing anywhere between 15 and 30% of that money that goes straight to Expedia.

You could attempt to haggle the hotel below $99 because it will actually get more money out of the arrangement and plus yes, If you have an issue and you need to cancel or something was wrong and you want compensation, it’s easier to deal directly with the hotel than with Expedia.

It’s just less convenient, you’ll have to call or email correspondence rather than just “booked it, packed it, ****ed off” - Peter Kay