How much of our technology is due to scientific advances done by SCIENTISTS?

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I have a question. You know how in physics the physicsts sometimes independently derive the math they need even though some mathematician before them has derived the whole theory already? I wonder how much of that is the case in general technological development.
It is said that science is needed for technology and is part of the main development of technology in the recent years. But is that true?
How much of this "science" is just the engineers tinkering with shit and seeing what sticks, what they like, and how much of it is impassionate research by people unconcerned with engineering doing "proper science" which just so happens to be applicable to engineering.
If you look at the ancient developments in engineering and technology they are quite impressive even by today's standards (e.g. pic related), and it doesn't seem to come from scientists but rather engineers.
It seems like most of science that is done by scientists as opposed to engineers is pure garbage and the claim that this stuff is what brought us to the moon and gave us the computer seems like a complete meme to me. But how actually true is it?