(Because once we do know how to do it, it gets a more specific name like "machine learning")
Since there isn't a plain sequence of steps that can be followed to explain the output, i'd say a different term is justified. Whether you call that "intelligence" is debatable.
There is a sequence of steps, it's just very long and unintelligible to humans.
I'm not even sure I disagree with you: Quantity has a quality of its own.
You wrote the code that tries random directions, but you are not choosing which directions it takes when executed.
> a precise rule (or set of rules) specifying how to solve some problem
The only difference is the rules are adjusted (trained) over time, rather than being written down once by a programmer.
It's a general trend in AI that:
1. Some problem in the AI-domain is solved with a new method (say, barcode scanning or handwriting recognition as historical examples).
2. This new technique is referred to as AI and not algorithmic.
3. Over time the ability of AI is pushed further.
4. At some point the method shifts from being understood as AI to being considered algorithmic.
It has to use/learn data within a defined scope, so even with dozens of algos, they are all well within defined ranges of variables.
Still within reason to be called well written software that performs the tasks that it was designed to accomplish.
When thinking about this it's a bit like if I submit an application on fiver to write code for me that takes me from A to Z. I get the code back, I don't understand it and I didn't write it, is it still an algorithm? All of the same can be applied to a neural network.
Is natural selection an algorithm?
Is how the universe works an algorithm?
I think it's useful in every day life to distinguish what humans do and what something out of human control does. It can also be useful to be a bit philosophical and lump definitions together, but this only works if everyone agrees that they're doing this in a discussion.