I love rhymed poetry and love writing it. Historically, much of the greatest poetry in the English language is rhymed, so I just laugh at the current restrictions and go on writing it! Unrhymed poetry with a consistent meter, such as you describe in Russian “white poetry” is known in the English tradition as “blank verse”. Some of Shakespeare’s plays are written in blank verse, and some, of course, are rhymed. So no one should be complaining if we go on writing English rhymed poetry, ha! ;-)
I love your descriptions of beating out the meter and using a rhyme generator. This is so close to what I do, though I can usually call up the rhyming words by just counting through the alphabet. One of my recent poems has this list across the bottom of the draft page:
aye by buy cry die fie defy fly hie lie my nigh pry ply wry rye sigh shy sky sly spy decry tie try vie
Bead breed bleed creed deed cede feed freed greed glede heed indeed lead meed need reed read seed screed steed treed teed tweed weed
So it’s the same thing, only as a native speaker it’s easier for me to think of the words. And occasionally I have to look elsewhere for rhymes myself.
Anyway, I loved reading through your piece!