I use Google flights and Kayak all the time, but truthfully have never once booked a flight through them for myself. Essentially, I like to use them as search engines to compare airlines and fares all in one place, and many of the people on here do the same.
Having said that, I've used Orbitz and Expedia a lot for work (not booking my own flights, but when I've had to book flights for people I work with or that we're contracting for various projects that require travel). I've found the big name, reputable 3rd party sites like Orbitz and Expedia to be reasonably reliable and can't say I've ever experienced a hassle with either that stands out all that much from the kinds of hassles I've dealt with from the airlines themselves - there can be change fees and restrictions and hoops to jump through on cancelations or changes, some customer service reps are go-getters eager to help out while others want to give you the runaround, sometimes your options are severely limited no matter who you book through because you booked the least flexible fare, etc. Mind you, it says something that I only trust my personal travel by booking direct with the airline.
What many people don't understand is airlines often have a best price guarantee - they want you to book directly through them, and want their website to be the destination that comes to mind when you're looking for a flight. Even if you can find a cheaper airfare listed on a 3rd party booking site, it is possible to call the airline and mention to them you're looking at a lower price on another site. They'll often honor that price, or even give you a voucher for the difference. The catch here is this usually has to occur within 24hrs of booking - which is to say, if you book with an airline and then see a lower price on Expedia or some other site within 24hrs, you can call the airline to get either a credit or voucher. Here's Delta's as an example (though I'm not sure American offers such a guarantee anymore):
https://www.delta.com/content/www/en_US/traveling-with-us/planning-a-trip/booking-information/online-booking/best-fare-guarantee.html
Third party sites can be great if and when everything about your travel goes well. You might have easily been able to shop around on a single site and found a good fare, and saved yourself some money on that wonderful trip. It's when things go wrong that they can sometimes be a bit of a hassle. This isn't to say booking directly with airlines assures you your experience will be hassle-free, and your experience with a 3rd party site is guaranteed to be a miserable one. Rather, it's more a case of how much you wanna stack the odds either for or against you - the more parties you involve in the process, the greater likelihood you may get the runaround or have to jump through hoops if a flight is canceled or delayed or connection missed or some other unexpected problem pops up. The more people wanting to make money off your transaction, the more fees you might come across. Again, nothing is ever really a sure thing, and there's always a chance you'll beat any odds or lose out no matter how good your hand. But ultimately your leverage is certainly higher when having booked directly with the airline.
