Magnetic cores can be used to do logic, and there were a few solid-state computers based on that premise (with Mhz clock rates long before transistors caught up)
It is possible to build the equivalent of an FPGA with cores and programmed jumpers like this. It could be useful on the surface of Venus if sufficiently high temperature materials were used.
Just to clarify, these TROS modules don't use core, but "plain old" transformers. The difference is that cores are stateful and nonlinear (they get magnetized), while transformers (more or less) just give you electricity out when you put electricity in.
As you said, magnetic cores were used for logic in some machines. Cores were also used in core memory, which was the main way of implementing RAM until semiconductor RAM came along. Cores were also used in the Apollo Guidance Computer's core rope ROM modules.
Ferromagnetic materials have a hysteresis, a stickyness to their field... materials with large hysteresis are used to make permanent magnets, and those with very low hysteresis are best for transformer cores, because each transition through hysteresis represents loss. All cores have it, it is an engineering optimization to choose a "hard" (high hysteresis) or "soft" magnetic material.
If you want to store a bit, as in core memory, or magnetic logic, you'd use a "hard" material. It is likely the cores in the TROS module were "soft".
If the hysteresis is low enough, even the normal atomic motion due to heat can cause the field to dissipate over time.
IIRC it's not just cores in general that do the computation, is it? Saturable reactors (https://en.wikipedia.org/wiki/Saturable_reactor) allow the use of magnetism for amplification, allowing one to build computers that would last a very very long time.
It is possible to build the equivalent of an FPGA with cores and programmed jumpers like this. It could be useful on the surface of Venus if sufficiently high temperature materials were used.