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The only thing that separates your mechanism for doing addition from what computers actually do is efficiency. Computers can only add numbers up to some fixed size, e.g. 64 bits, and you have to use repetition to add anything larger.

Does that mean computers are not "really doing" addition?

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There’s a lot more different than efficiency. We can program computers with algorithms capable of computing any possible addition, the limitation being only the memory of the computer and time, not the algorithm itself. Those algorithms are genuinely doing addition in a way that a pre-computed lookup table is not. It’s the difference between computing an addition in your head and just remembering that 2 + 2 is 4.

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