Right, there are algorithms to train and run neural nets. But the calculation that the neural net itself does can't really be described as an algorithm (unless you distort the term to lose all meaning).
They used to be called black box algorithms, if that is a salve.
Yes, neural networks are "constructs", not algorithms, we fully agree.
Fortunately, a combination of algorithms put together in a specified way is, in of itself, an algorithm. Most are various forms of addition ;-)