I think the answer to your question depends on how one should interpret the phrase flat tongue. So, let's try to read the crucial parts of the article, bearing in mind that our brain tends to save energy even when we talk.
First of all, as the article says, there are different kinds of retroflex consonants:
The Latin-derived word retroflex means "bent back"; some retroflex consonants are pronounced with the tongue fully curled back so that articulation involves the underside of the tongue tip (subapical). These sounds are sometimes described as "true" retroflex consonants. However, retroflexes are commonly taken to include other consonants having a similar place of articulation without such extreme curling of the tongue; these may be articulated with the tongue tip (apical) or the tongue blade (laminal).
Retroflex sounds articulated with the tip of the tongue (known as apical) are basically limited to plosive consonants, such as [ʈ], [ɖ] and [ɳ] or flaps, such as [ɽ] and [ɭ]. To be more specific, Wikipedia calls these sounds apical post-alveolar:
Apical post-alveolar, with a somewhat concave tongue (the tongue is curled). These occur, for example, in Hindi and other Indo-Aryan languages. (Hindi has no retroflex sibilants, although some of the other Indo-Aryan languages do.)
The tip is the most agile part of the tongue, so it can easily curl, but if the sound we are trying to make is continuous, then it could get tired pretty soon. Plosive consonants are articulated with a temporary occlusion followed by an immediate release of the air that comes from the lungs. Flaps also have a similar articulation. So, plosives and flaps are not difficult to pronounce even with the tongue completely curled. However, articulating a fricative or an affricate sound with this configuration of the tongue it's a different story. Fricatives and affricates consonants are continuous sounds, so eventually our tongue will get tired if our language has many of these kind of sounds in its phonetic inventory (Mandarin has four and uses them quite often).
That's why there is another group of retroflex sounds that have a slightly different articulation. They are pronounced with the blade of the tongue, which is that portion of the tongue just behind the tip. These are called laminal post-alveolar:
Laminal post-alveolar, with a flat tongue. These occur, for example, in Polish cz, sz, ż (rz), dż and Mandarin zh, ch, sh, r. (in IPA Mandarin zh, ch, sh, r are [ɖ̥ʐ~ʈʂ], [ʈʂʰ], [ʂ], [ʐ] respectively).
I think that in the article they used the phrase "flat tongue" to indicate that the consonant is pronounced with the blade which touches the alveolar ridge resulting flat to some extent. In fact, in this state the tongue itself is not completely curled, it only raises up a bit but the sound produced is still a retroflex.
The amount of energy involved in the articulation of laminal consonants is less than the amount required for the apical consonants, because now the tongue is more relaxed. Considering that articulating a fricative or an affricate sound using the tip of the tongue is really expensive in terms of energy, our brain developed the tendency to pronounce these kinds of consonants using the blade of the tongue instead of the tip.
I'm not saying that it's anatomically impossible to pronounce a fricative or an affricate sound with the tip but, if you try, you will notice that it's a little challenging and anyway the sound produced will be somehow different from the Mandarin equivalent. In the case of the fricatives [ʂ] and [ʐ] there is the risk that if you put too much force in the articulation with tip, the tongue will stick to the alveolar ridge and because of this obstacle the air will be released laterally resulting in a lateral retloflex sound, which is not what you want.
For this reason, in Chinese (like in all the other natural languages) fricatives (sh, r) and affricates (ch, zh) tend to be pronounced with the "flat tongue". In conclusion, it's a matter of terminology: retroflex doesn't necessarily mean that the tongue is curled back, it means that it is at least raised up.