Why adjoining non-Archimedean element doesn't work as calculus foundation? Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used in non-standard analysis. Why does this approach fail?

The construction you link to will work as far as it goes, if your only goal is to produce some non-Archimedian ordered field. However, the model-theoretic construction is much stronger; it will guarantee that every formal statement that is true about the standard reals is also true for the non-standard reals, as long as you replace all constants in the statement with their non-standard counterparts. Without this, it could be difficult to transfer results between the standard and the non-standard worlds.
For example, if one takes your simpler construction and adjoins a square root of $-1$, will the resulting field satisfy the Fundamental Theorem of Algebra? You would have to repeat the entire proof from the real case, if it goes through at all. With the heavy-caliber nonstandard analysis, you can just says that every nonconstant degree-$n$ polynomial over $\mathbb C$ has a root in $\mathbb C$, thus every nonconstant degree-$n$ polynomial over $\mathbb C^*$ has a root in $\mathbb C^*$. Case closed.
