What is the distinction between verbs like "to regard/concern" and prepositions like "regarding", "concerning", "about"?

"regard", "concern", "refer" are stative verbs that express the state of being related or connected to something.

Why is it that words like "regarding" and "concerning" are not considered verbs but prepositions. Are these prepositions essentially just irregular verb forms?

Here is a section from CGEL that directly addresses your question.

(Note that CGEL spends some time defending the claim that traditional grammar is wrong in treating the gerunds and the present participle as different verb forms, and says that instead we should speak of gerund-participles, but let's not dwell on that.)

2.3 Prepositions vs verbs

For the most part, verbs are clearly distinguishable from prepositions by their ability to occur as head of a main clause and to inflect for tense. There are, however, a number of prepositions that have arisen through the conversion of secondary, non-tensed, forms of verbs:

[21]  i  [Barring accidents,] they should be back today.
        ii  There are five of them [ counting/including the driver].
       iii  [Pertaining to the contract negotiations,] there is nothing to report.
       iv  [Given his age,] a shorter prison sentence is appropriate.

The basis for analysing the underlined words here as prepositions is that there is no understood subject. This is effectively the same criterion as we have used in distinguishing prepositions from adjectives: prepositions can be used in adjunct function without a predicand, i.e. an element of which they are understood to be predicated. The preposition counting in [ii], for example, is to be distinguished from the gerund-participial verb-form in:

[22]  [Counting his money before going to bed last night,] Max discovered that two
          $100 notes were missing.

The boundary between the prepositional construction [21] and the verbal [22] is slightly blurred by the usage illustrated in:

[23]  i  [Turning now to sales,] there are very optimistic signs.
         ii  [Bearing in mind the competitive environment,] this is a creditable result.
        iii  [Having said that,] it must be admitted that the new plan also has advantages.

These differ from [22] in that no subject for the underlined verb is recoverable from the matrix clause. They are similar to what prescriptivists call the 'dangling participle' construction illustrated in examples such as ^*Walking down the street, his hat fell off, ungrammatical in the sense where it was he, not his hat, that was walking down the street. Unlike the latter, however, the examples in [23] are generally regarded as acceptable. They differ from the prepositional construction in that there is still an understood subject roughly recoverable from the context as the speaker or the speaker and addressees together. Syntactically, they differ from the prepositions with respect to the dependents permitted: the verbs in [23i—ii], for example, accept the same dependents as in tensed constructions. In [i] we have the adjunct now as well as the complement to sales, and we could add other adjuncts such as briefly or if I may. Similarly, in [ii] the PP complement in mind is part of the idiom bear in mind, and again we could add adjuncts (e.g. bearing in mind, as we must, the competitive environment). In [iii] having is a form of the perfect auxiliary, and - unlike any preposition - takes an obligatory past-participial complement; little expansion is possible this time, but that is because having said that is a more or less fixed phrase in this use.

The main prepositions that are homonymous with the gerund-participle or past participle forms of verbs are as follows:

[24]  according † T       allowing F           barring †       concerning       counting
          excepting               excluding             failing †         following          including
          owing † T              pertaining T       regarding      respecting          saving †
          touching †            wanting †             given              gone † BrE        granted

The symbol '†' indicates that the preposition differs in complementation and/or meaning from current usage of the verb: we have prepositional according to Kim but not verbal ^*They accorded to Kim, and so on. Gone differs from given and granted in that the corresponding verb is not understood passively; it is used, in informal style, with expressions of time or age as complement: We stayed until gone midnight ("after"); He's gone 60 ("over").

As prepositions, the items in [24] take an obligatory complement - an NP, except for those marked T or F, which take a to or for phrase respectively. The prepositions during ,notwithstanding, and pending contain the •ing suffix, but are not homonymous with a verb.

There are also a few deverbal prepositions that take content clause complements: given, granted, provided, providing, seeing (see Ch. 11, §4.8).

Now let's consider these sentences you asked about:

[1]  a.  Dave was writing with a pencil.
       b.  Dave was writing using a pencil.

In both [1a] and [1b], the phrase in boldface functions a means and instrument adjunct. But that function is realized by phrases of different kinds in the two sentences.

In [1a], this function is realized by the preposition phrase (PP) with a pencil. Here the preposition with is the head of the PP (CGEL, pp. 673-675).

In [1b], this same function is realized by a non-finite gerund-participial clause.

There is nothing unusual in one and the same function being realized by phrases of different kinds. Take the function of predicative complement (PC). A PC can just easily be an adjective phrase (He is very handsome) or a noun phrase (He is a doctor). When it comes to adjuncts of manner and instrument, they are most frequently realized by PPs, but they can also be realized by certain other types of phrases.

The reason why using is a verb in [1b] is just the one given in the section from CGEL reproduced above: there is an understood subject, namely Dave.