Unless you're getting some great deal from booking through an agency vs. the airline itself, why do it? When problems come up it is MUCH easier to deal with the airline directly and not first have to go through some agency. Also know that not all available flights are listed on these sites. Call the airlines directly if you want more choices.

Why book thru an online agency? Because airline websites only sell their own flights + codeshares so if you need a more complex itinerary involving more than one airline, you will have no choice but to go thru an online agency.
Also, airlines such as Hawaiian do not take non US credit cards as payment so was impossible for me to purchase from their website. They wanted a US zip code. It might have changed since I last booked.
No airline wants you to call them direct to ask for flight choices when it is all online.
As for issues where the flights have changed, it is the airline that has initiated the changes for whatever reason so you are not going to get a different experience when the flight has changed to something you don't want all because you booked direct.
I book flights all the time; some thru airline websites if it is a simple journey and others thru agencies if the price is better.