The Oxford English Dictionary does list "intellect" as a verb. It's a fairly obvious coinage, in one sense: English adaptations of Latin verbs are usually formed from the Latin past participle (compare elect, eligible, both from Latin eligere, past participle electus).
It does not define it as meaning "ascertain the intelligibility of something." It just says " To understand or interpret by means of intellect. Now chiefly Philos."
There's a citation from 2006 that also features the uncommon related word "intellectible":
2006 F. E. Cranz Reorientations of Western Thought x. 5 "The forms
in matter..are not intellectibles or intellects in their own right;
they have become such only in the intellect which intellects them."
Whether or not this word should exist is a matter of opinion.
Words ending in -ible don't always have related verbs. For example, the adjective legible does not have a corresponding adjective "lect". We just have to use the English word "read." We also don't have a verb that comes from the Latin verb edere, the source of edible, although it is technically distantly related to the English verb "eat".