tone and inflection can be, to a certain point, represented by diacritics so that (jut for the sake of example) " shi " & " shî " could be told apart. You could even use a combination of diacritics and variant spelling to represent homonyms: " shi " , " shih " , " shee " .
While Chinese is tonal and could benefit from diacritics in a consistent manner, this doesn't apply to Japanese, which lacks such tones and inflections. The nearest thing to this is vowel length, which is already represented in most romanisation schemes, and still fails to distinguish between homonyms.
Another random example: the verb "oru" can mean any of:
exist
break
fold
pick (a flower etc.)
weave
And "sasu"...
insert
point
pour
shine
strike
Without kanji, or the extra context normally present in spoken communication, it's a lot harder to distinguish these. And teh pronunciation is not noticeably different for any of these homonyms, so any orthography differences will be very arbitrary.
Given that Japanese already had established orthographic rules by first contact with Europe, they simply wouldn't adopt for their own purposes an irregular scheme. While a foreign power might use an irregular scheme to write Japanese, that would not be adopted by the Japanese, unless Japan was essentially made into a puppet state first.
(I am actually fluent in Japanese)