Is there direct evidence that the tibialis anterior and peroneals are central to the short foot exercise ? Yes ,please see the link below to a recent study ( H Yoon et al 2017) .

The researchers took care to make sure that subjects activated the abductor hallucis in the exercise , so although the toes remained straight they would have have exerted pressure on the ground .

Maximum voluntary isometric contraction for the tibialis anterior and peroneus longus was recorded for each subject . Subsequent results showed an EMG in the peroneus longus of 46.3422.76 % of MVIC for flat footed individuals with the ankle plantarflexed , and of 51.4514.54 with the ankle dorsiflexed . See table 2 in the paper .

That level of activation is enough to produce a progressive strengthening response in the peroneus longus . In my opinion the tibialis posterior is likely activated to the same sort of level as the peroneus longus in the short foot exercise .

Provided the toes are kept straight and pressed into the ground during the exercise then the short foot exercise ticks a lot of boxes .

Link to paper
Comparison of the Foot Muscle EMG and Medial ... - ScienceCentral
by H Yoon - ‎2017 - ‎Related articlesThe short foot exercise (SFE) is effective in increasing the height of the MLA for people with flat feet. Most of the research related to the SFE has simply evaluated the efficiency of the exercise using enhanced ABH electromyography (EMG) activation.