Those are good links, but examine that last one:
SN 45.8 and AN 9.36 mention only jhana, and in fact SN 45.8 expressly defines sammasamadhi as the four jhanas. AN 9.36 says the ending of mental fermentations requires jhana.
AN 4.41 mentions jhana as one of four developments of concentration, for pleasant abiding here and now. The other developments of concentration would be interesting to investigate, but SN 45.8 isn't thereby superceded, nor is AN 9.36.
MN 117 simply mentions that the previous seven steps of the Noble Eightfold Path are requisites and conditions of sammasamadhi, the eighth step. This goes to my main point, that sammasati must be practiced before sammasamadhi - therefore, explaining sammasati as requiring sammasamadhi is incorrect.
Therefore, I don't see an exaggeration at all.
As an aside, the Dipa Sutta, which you linked first, exemplifies my point: someone desiring to attain jhana should practice anapanasati, which is to say all 16 steps. It doesn't say someone wishing to attain jhana should practice the first tetrad, and then after jhana practice the remaining tetrads. All four tetrads are practiced before jhana. The Sutta says so.