When talking about shi (and absence of si), to say "there is no si but shi in Japanese" is not really correct. The truth would rather be "there is no distinction between si and shi in Japanese".
In other words, there is only one such "voiceless sibilant" phoneme in Japanese, which is usually written as /s/, and さしすせそ are phonemically parsed as /sa si su se so/, all sharing the same consonant.
But this /s/ sound somewhat changes its color when it is followed by /i/ (or merged with /y/), under the influence of its palatal feature, resulting in a sound a bit more similar to English sh.
In English, there are two "voiceless sibilant" consonants, which are /s/ and /ʃ/ (s and sh), and speakers would naturally take it for granted that these two are different and independent from each other. To their ears, while consonants in さすせそ sounds almost identical to /s/, that of し feels more like /ʃ/, and the Hepburn romanization (de-facto standard romanization system for modern Japanese) spells consonants just according to these impressions.
So, to summarize, while English sounds s and sh are expressions of /s/ and /ʃ/ respectively, Japanese s and sh are "allophones" of one phoneme /s/, which are kind of expedientially written in two different ways according to the impressions conceived by English speakers.
In fact, Japanese sh sound is not completely identical to English sh and falls somewhere between English s and sh. I don't think there would be ANY problem if you pronounce し like "see" rather than "she" (or like russian сь in some cases like ～ました), at all.
The same story applies to chi and absence of tsi (despite tsu existing). The two consonants are phonemically the same (commonly written as /c/ avoiding writing in two letters), which means "there is no distinction between chi and tsi".
About distinction between t and ts/ch is another long story. All たちつてと are believed to have shared the same phoneme /t/ until recent period, despite that ち and つ no longer kept the original t sound for these several centuries.
However during the last century, hundreds of imported words from foreign languages (mainly English) urged people to begin distinguishing ち chi and てぃ ti (like in チー (a mahjong term) and ティー ("a golf tee")), which automatically meant two consonant phonemes involved within the two sounds, namely, /t/ for ti /ti/ and /c/ for chi /ci/.
Now たちつてと is generally considered to be /ta ci cu te to/ (some theories stick to more classcal /ta ti tu te to/, but they can't explain the distinction between チー and ティー very well). In this respect you can say that たちつてと contains two differents consonants that historically used to be one.