At the RTL or Moore-machine level, a computer consists of some register state (the flip-flops) which changes, at each clock transition, to a deterministic function of its previous state and its input lines at the time. So you need a way to compute the new state as some arbitrary Boolean function of the previous state.

The flip-flops each have a Q output and an inverted Q̄ output, which comes for free due to the symmetric construction of the flip-flop from a pair of vacuum tubes and other components. You can form any arbitrary Boolean function of the flip-flop state from a sum of products of the Q and Q̄ variables; it's a simple matter of applying De Morgan's theorem and distributivity. (Or, for that matter, you can use a product of sums by doing it in negative logic.) The flip-flops can supply not just all the memory but also all the inversion, signal level restoration, glitch elimination, and amplification, none of which can be done with diode logic.

So, indeed, diode combinational logic is not capable of computing arbitrary Boolean functions of its input lines. But it is indeed capable of computing arbitrary Boolean functions of the flip-flops' state. And that is all that is needed.

The LGP-30 manual gives a complete description of how the computation works at the Boolean equation level, a description which I have not yet managed to grok.

If you're interested in this kind of thing, you might be interested in notes/non-inverting-logic.html in Dercuano: http://canonical.org/~kragen/dercuano-20190711.tar.gz. (Like most things in Dercuano, it's unfinished.)