Rhymes – Can English Have Words That Are Both Alliterations and Also Rhyme?

This is a very deep question and I've wondered myself, often, why perfect rhymes sound so awful.
I don't have an answer (let alone the answer). All I have is some pieces.

Item: There is no doubt that such an effect exists, and is predictable and general.
It's similar to the priming that occurs with a ticking clock that jolts us when it stops.

Item: There is significant phonosemantic coherence among the 483 English rimes.
This can provide a semantic "rhyme" to match a phonological one in end rhyme.

Item: Rhyming poetry is a modern invention.
There is no known poetic tradition anywhere using end-rime before around 300 AD.

Item: Rhyming poetry reached its zenith in Medieval Latin religious and goliardic poetry,
leading directly to the rhymed Tuscan of Dante, and forgettable attempts at English rhymed epics.

Item: End-rhyme is significantly easier in a suffixal synthetic language, like Latin or Italian,
than in an uninflected analytic language, like English, as John Ciardi points out.

But that's just pieces. What they suggest to me is that there is a significant anticipation set up by a rhyme scheme, just like the anticipation of a clock's ticking that allows us to cancel it out automatically. Until it stops ticking and we're alerted by the unmet prediction. This has the same feel.

I suspect that the psychological effect of rhymed poetry is such that the pleasant effect is mediated by an expectation of a patterned phonological difference, which is not met by absolute phonological identity.

I also suspect that the difficulty of making rhymed poetry in English is a big part of the reason why it's fallen out of favor in official poetry. That, and the rise of popular music, which certainly has lots of uses for rhyme, but is not officially considered poetry, since a lot of people pay a lot of money for it.

I don't believe there are rules, but there is nomenclature and categorization. From Wikipedia:

A basic distinction is between rhyme schemes that apply to a single stanza, and those that continue their pattern throughout an entire poem (see chain rhyme). There are also more elaborate related forms, like the sestina – which requires repetition of exact words in a complex pattern.

In English, highly repetitive rhyme schemes are unusual. English has more vowel sounds than Italian, for example, meaning that such a scheme would be far more restrictive for an English writer than an Italian one, as there are fewer suitable words to match a given pattern. Even such schemes as the terza rima ("aba bcb cdc ded..."), used by Dante Alighieri in The Divine Comedy, have been considered too difficult for English.