Rather than "entrust", I think that a translation that fits better 委ねる in this case is "to devote oneself".
I think that you are quite there with your translation. Maybe would sound a bit better in this way:
In ancient Greece, philosophy, that is the research for the truth to which freemen devoted themselves, was purely an intellectual occupation [...].
It is true that 委ねられた is passive in this case, but I think that if instead you render that in English in an active way (preserving the original meaning of course) it sounds better and easier to understand. After all, you can say "I wear a t-shirt" and "the t-shirt is worn by me". Grammar (subject etc) is obviously different but the meaning is the same.
About how to translate, I think that that's the most interesting question here.
I will try to explain how I do. And please notice that this is just my way, not necessarily the best.
So, personally I always try to identify subject and verb first (maybe including some simple adjectives), and leave everything that is "color" for later.
In this case for example I look at: (古代ギリシャの)哲学が and then at 知的な営為だった.
So my first translation is: Philosophy (in ancient Greece), was an intellectual occupation.
Fine, makes sense so far. Then I add adverbs and other simple things. So I would look at it like: 古代ギリシャの哲学が、純粋に知的な営為だった.
Ok, so it was a purely intellectual business.
Finally I look at all that comes in the middle (that sometimes in Japanese is a lot, and very easily misleading).
So I take 自由市民に委ねられた真理の探究という. At this point I already identified that the main topic is philosophy. So I guess that that という is describing how we are defining philosophy in ancient Greece. 真理の探究 comes before という so we are talking of philosophy as the research of the truth.
There is only 自由市民に委ねられた left out, and at this point this obviously is there to explain who is involved (devoted/entrusted) with the research of the truth: the freemen.
After I broke everything down this way I look back at it and see if it makes sense as a whole using the following order:
Then look at what I called "the colors" above, trying to put them in a way that makes understanding easier. For example:
And finally put all together. At this point the translation should follow naturally and it's a matter of picking the right words (for example for 委ねる)
- 古代ギリシャの哲学が, 自由市民に委ねられた真理の探究という, 純粋に知的な営為だった.
...Now it would be very funny if after all this trouble it turns out that my translation is wrong! :D
EDIT: I think there was no すでに at the time of my answer. Maybe that comes from a later edit in the question. Anyway, it doesn't really change much everything I said above.