Can the mind be a finite automata and still invent the Turing machine?
Some Turing Machine Details


A Turing Machine does not have infinite tape, it has unbounded tape. There is no limit to how many symbols can be stored on it, but at any given time, only a finite number of symbols are ever on the tape. So while we cannot conceive of all runs of a machine, we can always conceive at least some states of the machine.
In a sense, Turing Machines carve out the "finite" portion of all languages, in the sense that it carves out the portion that can be finitely represented (by a Turing Machine). There are uncountably many languages, but countably many Turing Machines, so there's a bunch of stuff we will never understand because it is truly infinite, in too big of an infinity.
Most problems that we can conceive of don't actually require the power of a Turing Machine. Many can be written using primitive recursion or other well-founded recursion schemes. So most algorithms don't actually require the infinite-ness of Turing Machines.
For a Turing Machine that halts, there is always a function describing the max amount of memory used in relation to the input size. So we don't need to understand infinity, we simply need to understand the relationship.


Can a weak system invent a stronger one?

The answer here, I think, is yes, depending on your definition. For example, the language of all valid Turing Machines can be described using a context-free grammar, a computationally weaker formalization. You can even describe a regular language that describes what a valid Turing Machine looks like, and they're finite. (Spoiler alert, this doesn't say much, because you can map each integer onto a Turing Machine, so the regular language $\{0,1\}^*$ can be the set of all valid Turing Machines).

And, it's important to distinguish recognizing a thing, or inventing a description of something, and inventing all the things. We can recognize what a Turing machine is, but we can't see all of them, and because of things like the Halting problem, we certainly can't even understand all of them.

Other Infinite Things

This question is ultimately no different than asking how a finite mind could understand the natural numbers, or the function $y=x$ in $\mathbb{R}^2$. Both are infinite, but can be described in finite ways.

But, there are functions and sets and real number and such that are truly infinite, that can never be described by any possible spoken or written formula (because there are uncountably many reals, but countably many finite sentences in any human language.) And there are more true theorems than there are proofs, a la Gödel.

So we can describe infinity, and speak of its properties, perhaps without truly understanding it, at least without understanding every facet of it.