I wonder if the circuit could be modularized into a quantum ALU. Where many of the otherwise identical components in the original layout, couldn't just be arranged as operations time, over a reduced number of distinct components.

Most digital algorithms would explode in terms of hardware needed, for increasing N, if we didn't distribute that computation in time, as well as across elements.

In most quantum computer designs, gates are signals generated on demand at runtime rather than material deposited at fabrication time. In this regime, the concept of an ALU makes no sense. Instead of just sending pulses doing the exact gates you know need, why would you instead expand that short sequence into long sequences that emulate potentially applying every arithmetic operation to every input and then mux out the result you already knew you needed. It's a lot of extra work for the same final result.

A quantum ALU would also be substantially harder to design, because of the need to maintain reversibility. For example, every operation would have to run as slow as the slowest operation (or else the timing side channel would measure which operation occurred).

Is this the same idea as using universal Turing machines (CPUs that can execute software) rather than conventional fixed-function Turing machines (ASICs/FPGAs)?

unfortunately this is kind of fundamentally impossible. the whole power of quantum computation is that big quantum computers can do computation on a massive state space "for free". that benefit only exists if you have enough qbits to hold the state space.