Is there some specific reason or logic that came behind this, or was it simply because everybody thought it was the best representation for the sounds that create the words that use ゃゅぇょ?
I can’t answer how this orthography came to be adopted, but it does make sense to use the characters for syllables ending with /i/ before small letters like ゃ, ゅ, ょ, and sometimes ぇ.
The syllables represented by those combinations are palatalized sounds. The difference between か and きゃ, for example, is in the degree of palatalization in their consonants, and the consonants in /i/-ending syllables in Japanese are already palatalized compared to their counterparts before the other four vowels. As an example, き is transcribed in IPA as [kʲi] whereas か is simply [ka]. Here, the small letter [ʲ] indicates the preceding consonant [k] is palatalized. It means that its place of articulation is shifted closer to the palate. In this particular case, it is shifted forward from the velar, where [k] is usually articulated. The consonant in きゃ, きゅ, and きょ is the same as this palatalized sound, and they are indeed transcribed as [kʲa], [kʲɯ], and [kʲo], respectively.
The effect of palatalization is more obvious in し, ち and their voiced version じ (= ぢ). As you must have noticed, the consonants in these syllables are rather completely different from their counterparts in syllables ending with /a/. Indeed, they are represented by different IPA symbols, as listed below.
ざ [za] or [dza]
じ [ʑi] or [dʑi]
The symbol [ɕ] represents the voiceless alveolo-palatal fricative whose place of articulation is closer to the palate than that of [s], which is the voiceless alveolar fricative. [ʑ] is the voiced equivalent of [ɕ] and is similarly palatalized compared to [z]. In these cases, palatalization refers to a backward shift. And these are the consonants in しゃ, ちゃ, じゃ, etc.
じゃ [ʑa] or [dʑa]
Although the difference between the consonants in は and ひ may be less obvious, and that between な and に even less so, ひ and に are also palatalized.
The symbol [ç] represents the voiceless palatal fricative. It can occur in English words like “human,” too.
[ɲ] is the voiced palatal nasal, the sound for the Spanish letter ñ. I personally feel the degree of palatalization in に is not quite as much, though.
Besides all that about consonants, the semivowel [j] shares most of the features of the vowel [i]. In fact, や [ja] is like いあ [ia] pronounced in quick succession.
Therefore, it makes much more sense to transcribe the sound, say, [ɕa] as しゃ than さゃ, if that's what you are asking.