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The word cognize (cognise) does exist, but according to the OED etymology it seems a relatively recent word backformed from cognisant, cognition etc. and from the earlier recognise.
The relevant entry is:

Etymology: A comparatively modern word, formed with reference to
cognizance , cognizor , and the kindred words, and the earlier
recognize . It thus corresponds analogically, but not phonetically, to
Latin cognōscere , Old French conoistre , French connaître : compare
cognosce v. The prevalence of the ending -ize over -ise is apparently
due to the influence of the large class of verbs having etymological
-ize suffix.

†1. Law. (intr. or absol.) To take cognizance. 1659 T. Burton
Diary (1828) III. 129 The Judges..sit at Westminster, and they
cognize.

2. trans. To take cognizance of, take note of, notice, observe. 1821 Joseph the Book-man 107 [He] was cogniz'd by every eye.

1889 J. M. Robertson Ess. Crit. Method 72 Moved to incipient
hysteria where anon I should simply cognize pathos.

3. Philosophy. To know, perceive, become conscious of; to make (anything) an object of cognition.

a1856 W. Hamilton Lect. Metaphysics (1859) II. xxi. 19 It would
also be convenient..for psychological precision and emphasis, to use
the word to cognise in connection with its noun cognition.

a1856 W. Hamilton Lect. Metaphysics (1859) II. xxxvi. 329 They
first know,—they first cognise, the things and persons presented to
them.

1862 H. Spencer First Princ. i. iv. §24. 80 It is a material
object, and it is cognized by being recognized as such.

1876 St. G. Mivart Lessons from Nature vii. 196 We cognize an
object..by one act; we cognize that cognition by a very different act.

cognize

or cognise (ˈkɒɡnaɪz , kɒɡˈnaɪz )

verb
(transitive)

to perceive, become aware of, or know
^{Collins English Dictionary. Copyright © HarperCollins}

See also:

‘The distinction between intuitive and abstractive cognition does not depend on the object at all; the very same object can be cognized in both an intuitive and an abstractive way.’

(cognize on ODO)

Cognition, in the sense of "perception, sensation" is an old usage dating back to the 17th century:

In 17c. the meaning was extended to include perception and sensation.

from which:

the psychological processes by which individuals acquire and process information, generally applied to thought processes and memory. cognitive psychology the branch of psychology concerned with the study of cognition.

(Dictionary of Sport and Exercise Science and Medicine by Churchill Livingstone)

The verb is cognize. The originale term from which all other related term derive is cognizance, from Old French “conoistre” (to know). Note that it is a formal term.

"perceive, become conscious of," 1650s, back-formation from cognizance. The French word is connaître (Old French conoistre).

(Etymonline)

Firstly, cognite is not a word. There is no apparent entry of this word in any of the standard dictionaries. So, there is no question of it being a verb.

That said, you are right, that cognize or cognise does not fully cover the meaning of cognition because cognize is a verb of cognizance.

But, you can still use cognize as a verb for cognition on selected topics where cognition acts as a noun for cognizance, check below.

There are delineating similarities between Cognition (noun) and Cognizance (noun) which allows us to use "cognize" as a verb to cognition.

Following are the details:

So, whenever cognition acts as a noun to cognizance you can use "cognize" as verb to cognition.

Why, cogitate can't be used as verb for cognition?

Cognitize seems an over-complicated form, already used or not. Simply because someone has coined a form that apparently hadn't been called upon enough previously to evolve a popularly used form and a few people have picked it up doesn't mean it's the best fit. I cannot stand speaking of opinion that I cannot back up with sound reasoning, but for some reason - and I feel there is one, I've simply forgotten it, and am missing something that is preventing me from finding the support on the 'Net - it seems that the form sought here ought to be cognate, regardless of the fact that it is already in use for another idea.