Both sentences are ok using place or places.
When we wish to treat a number of countable units as a set, we can serve logic by changing them into a single collectivity if we introduce them as a plural noun:
Cinemas are the best place to take someone on a first date.
We can avoid this switch if we designate them as a singular thing initially (as subjects):
. . . [M]ore than ever before, the cinema is the place for people to meet and enjoy the big-screen experience . . . .
When we do use the plural subject, we actually think of cinemas as a plural countable noun that all sum together into one one conceptual place where something occurs.
If a thousand people (plural) can gather onto a crowd (singular), then so can all/most cinemas, schools, etc., gather into a singular concept when we wish to discuss that collection or set, even if we omit explicit reference to this "gathering".
In reality, then, in
Schools are the only place where . . .
the message is really
Schools are the only type of place where . . . .
Or, perhaps more accurately, something closer to:
"The set or network of locations called schools are the only type of place where . . . "
It could also be that we really have something like "type of place of all possible types of places."
If we leave out the words "set of" in "set of places", we can see how a plural word (places) remains, but represents a singular concept.
Notice how the normally removed or only understood language which I have supplied here contains singular nouns, showing that it is in fact a singular concept being used with the singular verb.
In any case, we have a slightly complex thing happening here. The underlying idea conceives of a singular idea (type, set, network, category, etc.), but some of that language is left out, and some of it might be somewhat transformed so that it appears we are making a subject-verb number agreement mistake.
I believe that the uses in your examples would be acceptable to most "educated" or "professional" English users or "experts". However, some people would recommend reformulating such language to avoid the potential problem in readers' minds.
The examples you gave would be much more easily and quickly forgiven in speech than in writing.
The idea under which we leave out certain elements of language where readers or listeners are expected to be able to understand that those elements are really there or understood to be there is called ellipsis, but if you will search for it be aware that it is not the same thing as ellipsis or ellipses which refer to the punctuation mark (. . . ).
Ellipsis causes a great deal of confusion to English learners both in understanding and producing language, because although it is extremely common in English, it is difficult (some would say impossible) to consciously learn where we acceptably can and cannot leave out grammatical elements.