More detail about how the causative and passive suffixes evolved.
The causative (-a)せる arose from just plain old す (su), itself the origin of the ubiquitous modern verb する (suru, "to do"). The original meaning was "to do [something]; to make XX do [something]" -- i.e., this served as a causativizing or transitivizing suffix.
Meanwhile, the passive (-a)れる similarly arose from just plain old る (ru). The original meaning was "[something] happens" -- i.e., this functioned as a passivizing or intransitivizing suffix.
In both cases, the root monosyllabic 助動詞 (jodōshi, auxiliary verb) had the classical Japanese conjugation pattern called 下二段 (shimo nidan, "lower two-step"). The "lower" part means the conjugated verb ending alternates between u and e, contrasting with 上 (kami, "upper") verbs that alternate between u and i. As Japanese developed, the root suffix was later conjugated into the 未然形 (mizenkei, incomplete form) of せ or れ, with the verbalizing suffix る added onto the end again.
For modern 五段 (godan, "five step") verbs (so called for the five different vowels that can appear on the end of the verb stem during conjugation), the old root form す can still persist for the causative, resulting in valid grammatical forms like いかす for いかせる, or のます for のませる.
(Note that sometimes the resulting causative form might sound identical to another verb, such as 置かす okasu "to make someone put something somewhere", which sounds identical to 犯す okasu "to violate, to rape". In such cases, the fuller causative form of 置かせる okaseru is probably preferable.)
As such, 行かす is not actually a contraction of 行かせる, and is instead the older form.
The causative and passive endings attach to the 未然形 (mizenkei, irrealis or incomplete form) of the verb stem.
For godan verbs, the mizenkei is the stem form ending in -a, such as 行か (ika) for 行く (iku), 積ま (tsuma) for 積む (tsumu), etc. The passive in classical Japanese was just る for yodan verbs (the precursor to modern godan verbs), developing later into れる, by the same mizenkei + る pattern described below for 出る (ideru). Similarly, the causative was just す, developing later into せる as mizenkei + す.
For nidan (only found in classical Japanese) and ichidan verbs, where the mizenkei stem is the same as the ren'yōkei (continuative) stem, the passive in classical Japanese was らる, developing later into られる. The causative was さす, developing later into させる.
As an aside, the す・る dichotomy for transitive / intransitive persists in numerous modern verb pairs, such as 出る (deru, to go out) and 出す (dasu, to put something out). These two verbs originate from the now-obsolete root verb 出づ (idzu, possibly idu in even older stages of the language). This verb was also a shimo nidan verb. The classical passive or explicitly intransitive form of a verb is [stem in mizenkei] + る. For idzu as a shimo nidan verb, this would be ide + ru, forming ideru. Over time, the initial i- dropped off, producing modern deru.
The causative or explicitly transitive form of idzu should be [stem in mizenkei] + す, or ide + su = idesu, but this pattern is slightly irregular and produced idasu instead, later shortening to dasu. (The shimo nidan causative / transitive pairs like this that I've looked at all take -eru and -asu, instead of the expected -eru and -*esu.)
Other transitive / intransitive pairs like this include:
- 増える / 増やす (fueru / fuyasu), "to grow or increase by itself" /"to make something grow or increase" (shimo nidan root)
- 溶ける / 溶かす (tokeru / tokasu), "to melt or dissolve by itself" / "to make something melt or dissolve" (shimo nidan root)
- 漬かる / 漬かす (tsukaru / tsukasu), "to be in a liquid; to become pickled" / "to stick something into a liquid; to pickle something" (godan root -- つく -- regular passive / causative formation, with idiomatic development over time)
TL;DR: Long story shorter, non-godan verbs cannot take the shorter causative form of す (su) or the shorter passive form of る (ru), due in part to historical development, and due in part to other related verb forms that already take a bare す or る.