From the article you linked to. Emphasis added.
It's a relatively close battle in the so important brightness department between the LCD screens that we have here, but it can easily be seen that the humble 4" IPS panel of the iPhone 5 has a slight edge above the others. Being able to produce brightness of more than 500 nits, the iPhone 5 really is the champion here when it comes to outdoor visibility. Of course, you'd rarely want to view you display at such a brightness indoors when there's not that much ambient light present, but we all know how important it is to be able to read your handset's display easily when out and about.
Thankfully, the Z Ultra, the One and the G2 aren't far behind, as their maximum brightness of 400+ nits allows the to achieve some pretty good visibility as well. The Super AMOLED-powered Galaxy S4 is noticeably worse in this respect, which is to be expected with the significantly lower brightness output of AMOLED screens. While the GS4 isn't really unreadable in bright daylight, it does appear dimmer than the other four.
So it's slightly worse than the iPhone for brightnss, but not once did they complain about it saying it has pretty good visibility. The S4 is clearly inferior here due to the SAMOLED, and that hasn't hurt sales one bit.
I agree that color profiles should be more accurate, but once again the most inaccurate screen, the S4 hasn't seen too many complaints either. Another thing to note was that most review samples aren't running the latest software build. I seem to remember reading another review where the color accuracy was exemplary, although customizable and had other settings, like a cooler setting to mimic SAMOLED type displays (why anyone would want this ???). So this was just one review, and it was quite favorable.
As for flagship/top dog, you can pretty much say that the iPhone isn't the top dog in most every category, it just does many things very well. The combined whole is more important that the individual parts.
What I get out of that article is that the screen is one of the best available. It may not be the best in every category, but it's pretty darn good. So out of the 300 phones on the market right now, this screen comes in 2nd or 3rd. That's not too shaby if you ask me. I just disagree with the premise that the screen is "average".
All in all, it'd be extremely hard if we have to choose an overall winner. Each display tends to be a bit better then other four in some areas, and worse in others. Combine that with the fact that all of them are actually amazing (with some small exceptions), and it gets even harder to say who's the best.