To add to the previous answer, there is no clear-cut distinction between 'regular' and 'irregular'; also, irregularities can often be explained and may hint at an old form or conjugational system. Eg strong verbs in English and German (ablaut conjugation), be-was-is-am (merger of different verbs).
As for ある, while *あらない is not used in modern Japanese, grammar (conjugational patterns) would predict this form and I would not be surprised to find it in some dialect. Also, we can negate ある via the 未然形: 非【あ】らず.
This slight irregularity in ある is considered insignificant enough to be classified as a 'regular' verb, but never mind by which name we choose to call it. There are also other slight irregularities: いらっしゃイます (not いらっしゃります), 乞ウた (not こった), 行ッた(not いいた).
The origin of these different conjugation paradigms is unclear. According to one hypothesis, many 一段 and 二段 verbs were formed by processes such as adding ある, 得る; and that there is an older conjugation paradigm explaining irregular forms.
Without going into details, observe that 3/4 of all verbs in Old Japanese are 四段; and a few 'irregular' words such 死ぬ・死ぬる・死に・死な(in the order 終・体・連・未 base) hint at an older 四段-like conjugation with a 連体形 ending in る.
See 'Bjarke Frellesvig and John Whitman, 2008, Proto-Japanese: Issues and Prospects' for a technical discussion (cited in his book 'A history of the Japanese language', I have not read it.)
See https://archive.org/stream/historicalgramma00sansuoft#page/150/mode/2up for a dated and less-technical discussion.
To summarize, it seems likely that your observations are not coincidence and that there is a relation between the different conjugation paradigms, but the details may be lost in the mist of time.