Strictly speaking, かつての遊び has only one corresponding verb, 担っていた.
かつての遊びは機能を担っていた。
Playing in the past used to take on the function.
And everything else is a relative clause that modifies 機能.
自己と他者に対する信頼感を育てる機能
the function of growing confidence in self and others
The tricky part is the word order. The relative clause is split into two and the topic of the main clause (かつての遊びは) is suddenly inserted between them. Still, since は is not usually used in a relative clause, this sentence is not really confusing to me. Usually this sentence is written like the following, which I think is much easier to understand:
- かつての遊びは、自己と他者に対する信頼感を育てる機能を担っていた。
- 自己と他者に対する信頼感を育てる機能を、かつての遊びは担っていた。
Here's another example of tricky word order:
彼女のことが、彼は嫌いだと言ったのです。
= 彼は彼女のことが嫌いだと言ったのです。
= 彼女のことが嫌いだと彼は言ったのです。
He said he hates her.
In the first sentence, the topic (彼は) is mentioned in the middle of the quote, and this is still a valid Japanese sentence, though uncommon. The three versions mean the same thing, but the first one looks a little more impressive to me because of its unusual word order.