Vsauce mentions a study by Joan Serra showed contemporary pop music is less complex in its musical motives than it was, which he explains by a wider diversity in form and genres.
The problem with studies like this comes down to definitions. How do you define 'pop' music in a way that makes meaningful comparisons across time possible? How do you define 'complexity' in a way that is consistent across different musical genres, which focus on completely different types of musical elements?
The study in question seems to be this one. As far as I can tell, the researcher in question is just using songs that are in the million-song dataset - but I can't see in the study where he gives any reasoning as to why this is a scientifically-consistent collection of music to compare across time. (this article has some of the same objections about the study.)
The researcher has also been limited to the particular data fields already available in the million-song dataset; They don't seem to have independently validated the usefulness of those fields, although perhaps that validation is available elsewhere.
To me, it also seems like classical rock used more rhymes, or more intricate rhymes than the contemporary pop music. Is this true? If so, why?
To find out if it's true, you'd have to come up with a formal metric for 'rhyme intricacy', and do the analysis. I'd be incredibly surprised if a lot of current hip-hop oriented pop music didn't score very highly on some possible metrics for rhyme intricacy. But ultimately the metric would be subjective - as would the definitions for "classical rock" and "contemporary pop".
Edit, following the definition of rhyme being added to the question:
I am not aware of any comparative studies of rhyme frequency. My guess would be that you would see a wider range in rhyme frequency in contemporary chart music compared to the chart music of 60 years ago. On one hand, there is a prevalence of dance-influenced music that often treats lyrical content as somewhat textural or incidental, rather than focal. On the other hand, the last 40 years have seen the onset of hip-hop music which is likely to be much higher in rhyme density (probably in any type of rhyme) than any sung music. In fact two of the studies I can find on measurement of rhyming metrics focus on hip-hop:
Using Automated Rhyme Detection to
Characterize Rhyming Style in Rap Music
Automatic Detection of Internal and Imperfect Rhymes in Rap Lyrics
Of course defining what a rhyme is still doesn't give an entirely unambiguous metric for measuring rhyme frequency or richness. Do we do it as a metric per second (which would be affected by tempo), or per word? Do we count repetitions of rhyming couplets? How exactly do we define what "the stressed vowel sound" is, when this can vary according to accent/locality? Are we interested in rhymes within lines, or only at line endings?
Taking a step back, focusing on the definition of 'good rhyme' might be at odds with what many artists are actually trying to achieve. The Wikipedia entry on imperfect rhyme explains why there are a number of creative reasons why it is desirable to use imperfect rhymes (or half rhyme / slant rhyme):
This can be used to avoid rhyming clichés (e.g. rhyming "knowledge"
with "college") or obvious rhymes, and gives the writer greater
freedom and flexibility in forming lines of verse. Additionally, some
words have no perfect rhyme in English, necessitating the use of slant
rhyme. The use of half rhyme may also enable the construction of
longer multisyllabic rhymes than otherwise possible.
Given that the creative use of imperfect rhyme can be seen as a good thing, it may be that the use of 'good' rhyme might subjectively be seem as naïve and clichéd.