No.
Godel's completeness theorem can not be understood without bringing in first order logic, because it is a statement of the expressitivity of the language(relative to its semantics). Other more expressive languages, like second order logic (with its usual semantics) is not complete. Trying to explain Godel's completeness theorem without bringing in the language is a path to confusion.
And your explanation of the first incompleteness theorem is also at best confusing. I must preface this with the comment that your definition of a 'theorem' matches what is usually called a sentence or a statement, and a theorem is usually reserved for a sentence which is proven by a axiomatic system. If the axiomatic system is sound, all theorems will be true in all models. The question of completeness is whether or not all truths(aka sentences true in all models) can be proven(aka they are theorems). With this more common usage of the words, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of the theory (that is they cannot be proved inside the theory).
Your description of the first incompleteness theorem is also true for complete logics, even for propositional logic (with your definition of 'theorem' as actually meaning statement). It has statements which is true in some models and false in others. This does not make it incomplete.
Actually, I think your statement muddies the waters and the parent gives clearer picture for those looking for simple statement of what's going on. The background is the fellow comment: https://news.ycombinator.com/item?id=48224739. Godel's simplest (and roughly original) statement is any system of axioms strong enough to encode arithmetic is either consistent or complete. You can "Or the set of axioms is not enumerable" (as in Second Order Logic and other systems). But when one says that, one has jumped from would normally recognized as logic (finite axioms and process) to a mathematical construct with some similarities to naive logic but which "my gran pappy" would see as logic.
I mean, you could formally construct an axiom system defined to choose (via axiom of choice) and assign a truth value to each of the independent propositions of first order arithmetic logic. There, consistent and complete system but not one that's a whit closer to being in usable by anyone.
I'm not even a finitist but I think being clear what's going on with these claims is important. It's like saying "the halting problem can't be solved by finite computers but my infinite-foo hypothetical computer can solve it, gotten mention that..."