The overall discussion in this text is contrasting different definitions of greatness and insignificance. Jesus flips the definitions on their heads.
But first, the main argument against the OP's suggested translation--entrusted to me(unto me)--is that the Greek can't mean that.
τῶν πιστευόντων εἰς ἐμέ...
has a very straightforward translation of:
those who believe in me...
After some analysis, I suggest a better translation (I'll explain in a moment) would be:
--the ones who put their faith in me...
First, the participle is in the genitive case. This case is the "kind of" case, or "in the class of". In other words, it's characterizing the noun it modifies by saying that noun is to be characterized by "believing".
Second, it's articular (has the article). So, the participle is functioning like a noun. If it weren't for the prepositional phrase, a good English translation would be the simple, "the believers".
So, the people referred to, while being called "little" and contextually associated with children, are modified by the participle to be in the class of those who believe in Christ Jesus.
Third, the preposition εἰς indicates the verb's action in some sense moves into the object of the preposition. This is different, in a small way, from ἐν which places the verb's action in the object. Thus my suggested translation of "put their faith in...". The resulting difference in sentence meaning between εἰς and ἐν is trivial in this case.
Fourth, the immediately preceding phrase is, ἕνα τῶν μικρῶν τούτων ("one of these little ones"). These words, except for ἕνα, are in the genitive case. So, this is a class of people, the little ones class. It also means that the participle phrase further modifies the "little ones" phrase. This is normally taken to be epexegetic (that is, it explains what goes before). In other words, Jesus is referring to people who are of the class little ones who are also of the class of believers. They are one and the same group characterized in two different ways.
This fits the context well since the discussion is about greatness. Verse 34:
But they were silent, for on the way they had argued with one another about who was the greatest.
But, Jesus defines greatness in terms of insignificance and service to others (verse 35):
"If anyone wants to be first, he must be last of all and servant of all.”
Children are considered as last. And the action of service is also brought out in verse 41.
whoever gives you a cup of water because you bear Christ’s name will never lose his reward.
And, fifth, the participle is in the active voice. So, it's not that the children are passively entrusted to Jesus, or in the middle voice which would mean that they entrusted themselves to Jesus. It's that the people are actively believing in Christ Jesus.
I think one could argue that the active sense coupled with the preposition εἰς (into) could mean an "active entrusting".
However, even then, I think we'd have something like:
...one of these little ones who are entrusting themselves to me by faith...
Lastly, I take the word σκανδαλίζω (typically, cause to stumble) to be a strong word. I think of it as to ensnare. The suffix -ιζω is a lot like English '-fy', in that it indicates 'to cause' or 'to make happen'. For example, our word 'pacify' means 'to make passive'. So, in the case of σκανδαλίζω it's "to cause to be trapped in a snare".
So, Jesus confronts "great" people here. If you who are great trap insignificant people so that they have no choice but to morally fail, then it is going to go very, very badly for you. In other words, we all know people who think they are the only "adult" in the room and who try to force all of the "children" to do their will. These so-called great people, when they cross a line, are in very deep trouble. That's the point of this text. It's actually a very good text for government officials to take to heart. It justifies the true meaning of "public servant" and strongly denies any parody of that term.
But, specifically focused on your question, you can't make the Greek say what you want it to say. Please don't think I'm attempting to dissuade you from your empathy for children. It's just that the text has more to do with those who have entrusted themselves to Christ then referring to specifically and only children.