You have to understand that color temp is not equal to spectrum ...
Different color temps will have different spectrums => true
Same color temps
=/= same spectrum
Just because you end up with the same color temp as natural daylight does NOT mean you will have the same results and spectrum as natural daylight ...
You could take a 3500K spectrum, strip away only some of the red part of the spectrum away, and obtain 5000K or 5780K ... Just like you could add blue, or cyan for eg.
=> There is an infinite number of different possible spectrums that would be labeled 5787K ... some extremely different from one another, some very similar ...
Think of color temps as the overall "tint" of a (wide) spectrum as perceived by the human eye ... you got it ... plants don't really care about K TEMP either ..