Infinite Boolean circuits as a model of computation
Your idea is equivalent in power to non-uniform circuits.  Thus, it doesn't formalize the concept you want it to formalize.  You were hoping that your notion of an infinite circuit would capture the power of uniform circuits / Turing machines, but it doesn't -- like non-uniform circuits, it is "too powerful" (more powerful than any uniform model).  Thus an infinite circuit is essentially still a non-uniform model of computation, even if it doesn't look like it on the surface.

Consider the following infinite circuit $C$: if the $i$th bit of the input is 1 and the $i+1$st bit is 0, then the $i$th output bit is $c_i$, otherwise that output bit is 0.  Here $c_i$ is a constant that is 1 if the $i$th Turing machine (in some canonical enumeration of all Turing machines) halts, or 0 otherwise.

This circuit is an infinite circuit, in your sense.  You can think of it as a sequence of circuits $C_1,C_2,\dots$, where $C_N$ is the circuit that takes $N$ bits of input and produces $N$ bits of output, obtained by zero-padding its input and feeding that to $C$.  Thus that sequence of circuits "converges" to $C$.

However, $C$ has the following peculiar property: given $C$, you can solve the halting problem.  In particular, if you feed in $i$ in unary as input to $C$, then the output will indicate whether the $i$th Turing machine halts or not.  Thus, infinite circuits can solve undecidable problems.  Like non-uniform circuits, infinite circuits are in some sense "too powerful" -- they're not realistic.

So your construction doesn't really eliminate any of the issues with the non-uniform circuit model.  It just recreates them, in a slightly different guise.



A separate challenge with your infinite circuit idea is that it is not clear how to handle circuits that produce a single bit of output.  Normally, in circuit complexity, we study circuits that take $N$ bits of input and produce a single bit of output.  It's not clear how such a sequence of circuits can "converge" to a single infinite circuit (what is its single output bit, and how is it computed?).