As you now describe it, it cannot be implemented (physically realized) at all. Or, conversely, any physically implementable refinement of it (which will not exhibit the entire range of behaviors) will be simulatable by a TM.

There are many abstract machines that cannot be implemented by TMs -- e.g. various oracle TMs. There is nothing special, surprising or new about that.

There are many formalisms that are more or less convenient abstractions for various kinds of systems. There is nothing special, surprising or new about that, either. In fact, some formalisms that can describe non-computable behaviors are commonly used to specify either software or hardware as they're convenient (like Lamport's TLA).

But you're making a claim about the Church-Turing thesis, which, as commonly interpreted today (as the physical Church-Turing thesis), is the claim that any mechanism that can be implemented by a physical mechanism can be simulated by a TM. Unless you show how to build a physical system that cannot be simulated by a TM, your claim is no refutation of the thesis; it has nothing to do with it. Your claim that arbiters in digital circuits cannot be simulated has not been established and is not recognized by scientific consensus.

> However, just as there can be an arbitrarily long amount of time between two steps of a computation, there can be a arbitrarily long amount of time for a message to be delivered.

This is a completely different use of "arbitrary". In TMs, the fact that an arbitrary amount of time can pass between steps means that any device, with any finite amount of time between steps, can produce the full range of TM behaviors. In your actor case, to get non-computable behavior, you need to show that the device can delay the message by every amount of time. You need to show that such a physical device can exist.

Put simply, it's one thing to propose a non-computable abstraction that's convenient to model some systems, and another thing altogether to claim that there are realizable physical systems that cannot be simulated by a Turing machine. The former is useful but mundane (in fact, all of classical analysis falls in this category); the latter has not been achieved to date.