"hanasimasu" is not exactly wrong; there is a romanization system in which "si" is how you write "shi".
If you want to invent scheme for understanding conjugation which works by cracking the romanized versions of words to create a pseudo-stem that could not actually exist in spoken language, it behooves you to adopt "si" and "ti", because they bring in a consistency needed by such a system to be complete.
> If you want to invent scheme for understanding conjugation which works by cracking the romanized versions of words to create a pseudo-stem that could not actually exist in spoken language
That's how all conjugation schemes work. There's nothing weird about this. Stems aren't supposed to exist in the spoken language. But they are observable in the spoken language.
Compare how a modern dictionary will give you ποιέω, a full and fully-inflected word which doesn't actually exist in ancient Greek, as the first principle part of that verb. This is done because the stem of the verb is ποιε-, and the epsilon ending the stem can be easily observed by its effect on most of the conjugational endings. It doesn't happen to affect the first-person singular ending -ω (to be precise, the contraction of ε- with -ω is -ω), so the dictionary form is synthetic, chosen to be informative.
I should have said that they don't exist in the written language; but using an alternative one lets them be written. This is why romaji appears handy in this matter. It lets us have a stem ending in "r", which doesn't change when we have "a", "u", etc after it.
On the matter of 'si' and 'ti', they are not appropriate in a model of modern Japanese, which recognizes e.g. the onset of ち as a phoneme distinct from the onset of と. The orthography uses a digraph / diacritic for this, writing "ちゅ" for the single syllable that belongs in the -u column of the row containing ち, but that's not a reflection of Japanese phonology; it's a historical artifact.
It's still the case that Japanese speakers have difficulty producing the hypothetical sound 'si', but that doesn't mean that the syllable which is notionally given that place in the kana table represents that hypothetical sound. In English we have the very similar rule that the cluster /sj/ may be reduced to /ʃ/, but this obviously doesn't prove that the word "sheep" begins with the phoneme /s/.
> On the matter of 'si' and 'ti', they are not appropriate in a model of modern Japanese
No romanization scheme captures all the phonetic nuances of Japanese.
And neither does the Japanese writing system.
They are not intended to be detailed models of the spoken language.
> that doesn't mean that the syllable which is notionally given that place in the kana table represents that hypothetical sound
It's all convention. The umbrella handle し is also "notionally present" in the table, and represents the sound only by convention.
I have a native language in which the written combination "ni" often, but not always corresponds to a palatized n, very similar to the Japanese one. In other situations, the palatized n has to be explicitly annotated as ň. There are also exceptional siguations, like the names Niagara or Nikaragua, or the word nikotín.
If we were not to have any conventions like this, we would have to write using IPA symbols! That has downsides. One is the proliferation of symbols. The other is the need to adjust the phonetic spellings for regional dialects, and over time as phonetics changes. In other words, the writing system being a detailed model of phonetics is not necessarily a good thing.
> In other words, the writing system being a detailed model of phonetics is not necessarily a good thing.
> No romanization scheme captures all the phonetic nuances of Japanese.
The fact that alveolar and palatalized sibilants both exist as contrasting phonemes is not a "nuance". It will be represented in every writing system that anyone ever puts forward, as indeed it already is.
The only advantage of putting 'si' in a Romanization of Japanese is that it corresponds well to the official alphabetical order of Japan. There is no other reason you'd do it.
The linguistic article I linked to at the bottom directly deals with that so I invite you to check it out. Yes, romaji with a color-coded hole may be a plebe’s choice to render the stem, but the stem itself is a reality of the spoken language and how it evolved, whatever way we choose to render it.
Because Japanese does not have both "shi" and "si", and does not have both "tchi" and "ti", the latter are available as a notation to represent the former, and have been so used.
There is no need to encode rules like s* + i -> shi, because that is taken care of by the existing understanding that si encodes shi.
The romaji notation is not phoentically accurate to begin with (and neither are the Japanese writing systems); like it doesn't capture nasalization of G followed by N and what not. The "n" in "na" and oni" is different, yet we don't write ñi or whatever to indicate the palatization. The Hepburn romanization is just based on what is or is not convenient relative to English.