many of our educational system aspects
Why would many aspects of our educational system be preferred?
Primarily because you're turning the noun-phrase "educational system" into an adjectival modifier, so that in effect you're "casting" noun-phrase {adjective-noun} into an adjective:
( (educational system) as adjective) aspects
That may be easier for the non-native speaker who is unsure of which preposition to use in a prepositional phrase, but the following is more idiomatic:
many aspects { of our educational system }
just as "facets of the diamond" would be better than "diamond facets".
The facets are not "diamond".
The aspects are not "educational system".
Why is it better? Because the noun-as-adjective is not the clearest and simplest way to represent the idea of "belonging to" or "pertaining to". The best way to represent that idea is to use a possessive form.
the diamond's facets
the facets of the diamond
the system's aspects
aspects of the system
When choosing between the two forms of the possessive ( system's or of the system) a rule of thumb is to use "of the {noun}" with abstract nouns and the genitive possessive with concrete (and personal) nouns. System is an abstraction, so aspects of the system is a shade better than the system's aspects. Diamond is concrete, so the diamond's facets is a shade better than facets of the diamond (if you have a particular diamond in mind and are not speaking of diamonds generically).
In that last recommendation, we are moving away from strict notions of what is grammatical to a more nuanced sense of what is euphonious. There is not much difference between these two versions,but I think the second version is the better one.
The software's worst feature is its ribbon interface.
The worst feature of the software is its ribbon interface.
The worst trait of Tom is his stubbornness.
Tom's worst trait is his stubbornness.