1. I've never heard anybody say that air carrier websites are always cheaper than third party sites. However, the savings can be minimal and when they are significant, it's often because the third-party site is selling bulk or consolidator tickets and is not acting as a TA.
2. It is true that CS in IRROPS and other situations can be extremely difficult, time-consuming, expensive and worst, handled as a courtesy, not because the carrier is required to. This means that if there are 400 pax to rebook, you may be the 400th.
3. The third-party site's T&C are in addition to the air carrier's T&C for fare basis and COC. This can mean that "non-refundable" or "non-changeable" really does mean that. In other words, if you don't fly exactly as scheduled, you lose 100% of the value of the ticket. Change fees can be much higher and flexibility may be very hard.
4. Even if you book through a good 3rd party site, you need to confirm with each carrier that you actually are ticketed on the specific flights the website says you are. A printed email from a third-parrty site won't get you a BP if it hasn't been properly ticketed.
5. There are indeed great TA's who can help you with complex itineraries and hotel bookings. They are well worth the fees they charge if you are not personally experienced with the destination.
Remember, if a deal is too good to be true, it's too good to be true.