I've only dealt with a few websites.
CO's is so straightforward, and has large icons to access the information the vast majority of users want.
But when you bring up schedules' CO' site will suggest the most convoluted, ridiculous routing arrangements. Someone not observant might book a flight from CLE to PHX with two connections. I'm not kidding.
When you first go to UA's site, you're presented with lots of choices of lines of text you can click on to get whatever. That's O.K. but it seems to follow the philosophy that being the world's largest airline, they must offer a bewildering array of "stuff" at their website instead of just the "basics." Also if you check on a flight's status at UA's site, it confirms that you're entering a secure connection, while CO's site just shows the flights status without establishing a secure connection. UA I guess feels that divulging whether flight #135 is on time or not needs to be on a secure connection.
TW's site is just plain stupid. In their roster of TW cities you access in order to check schedules, they list cities serviced by their code share partners. This is not unusual I know. But, when you attempt to access the schedule between those cities it tells you it can't find any service. For example at TW's site, try checking between Kuwait City and JFK (Kuwait is on their city list choices). It'll tell you "no service found." Maybe other sites do this too, I don't know.
Vanguard's site actually lets you pick the fare category, which is ridiculous cause most people just want the cheapest fare available and don't care if it's a 'K' or 'M' fare. It also makes it a little confusinf if you don't know what you're doing.
I haven't done a formal study, just sharing some observations!!!!!!!!