I will risk doing something I normally don't--answering a question that I'm not quite sure that I know the answer, but I think it might be useful for the OP and other learners, so please be gentle with me a little this time.
(This doesn't mean that I think other answers are wrong. What I really want to do is to simplify the problem, and make the reading obvious.)
Children can look at a variety of pre-screened content online, but can’t search for specific topics, as they would for homework.
Should we read it as "Children can look at a variety of pre-screened content online, as they would for homework.", and take "but can’t search for specific topics" as something we can omit?
I think not. Why? Because, usually, we don't insert our 'after thoughts' in front of a conjunction.
Why so? Because it will confuse the reader (or the listener).
Your case is different from, say, "I like Jane--I told you before, right?--as much as John." (Because in this example, as is not a conjunction.)
So, how should we understand a sentence with multiple conjunctions?
My answer is we should read it sequentially; and make sense of the sentence by using the nearest part possible. This is essentially the same thing StoneyB wrote about. It is also exactly what we do when we solve which pronouns refer to which nouns1. Omissions (or ellipses) also work this way. Most of the times, it refers back, and the rest of the times, it refers forward.
1) For example, "We keep the ice-cream machine in the spare room. This is mainly used by children, incidentally."
"Now what do you think about this? I thought I'd get a job in Spain for six months, and then ..."
(Examples were taken from Practical English Usage by Michael Swan, entry 590.)
To demonstrate how to read your example sentence, here is how I read it:
Children can look at a variety of pre-screened content online,
but [children] can’t search for specific topics,
as they would [search for specific topics] for homework.
The parts in brackets [...] were inserted to make the meanings clear.
This demonstrates two things:
- I read it sequentially, and
- I solved each missing part by using the nearest part possible.