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You said...
"If a/bc = a/(bc), then a/bc = a/b/c
Which seems nonsensical to me."
In fact that is completely correct  just put some numbers in...
1÷2÷3=(1/2)÷3=1/6=1/(2*3)
In the same way that distributing a minus over a bracketed term changes any + to a , distributing a ÷ over any bracketed term changes a x to a ÷.
This is a wellknown property in Maths which is covered when teaching multiplying and dividing fractions.
Someone once asked me "what is a/b/c/d?". Easy, a/bcd