Besides, considering morphemes as semantic often results in a completely different meaning than we actually intend. We aren't trying to train a chatbot to speak in prefixes and suffixes, we're trying to train a chatbot to speak in natural language, even if it is encoded to latin script before output.
(The point about semantics is also technically wrong. You would first need to specify your view of semantic compositionality before such a point can be evaluated, but the usual views of semantics don't have any such consequence.)
Sure, if you define "morpheme" as a collection of syllables that's meaningful to people using alphabetic script. I don't see any benefit to this compared to working with syllables directly, which is a meaningful concept regardless of the script used to encode them.
Cats, as noted, has two morphemes, despite having only one syllable. Syllables and morphemes are largely orthogonal, morphemes can be less than, equal to, or more than a syllable (and even when more than, may or may not start or end on a syllable boundary.)
(Also, syllables aren’t the minimal semantic units even of spoken speech, those are phonemes – a syllable consists of at least one phoneme, potentially more. But morphemes, even an alphabetic script if it isn’t perfectly phonetic, still don’t necessarily map to one or more phonemes, since is textual semantic unit may have no effect on pronunciation.)
Characters are not the semantic components of words—these are syllables. Generally speaking, anyway. I've got to imagine this approach would yield higher quality results than the roman alphabet. I'm curious if this could be tested by just looking at how LLMs handle English vs Chinese.