Can one prove the elementary divisor theorem for PIDs by elementary matrix operations?
According to Wikipedia, examples of PIDs with nontrivial $SK_1$ have been first given by Ischebeck in "Hauptidealringe mit nichttrivialer $SK_1$-Gruppe" (Arch. Math. 35 (1980), 138–139) and by Grayson in "$SK_1$ of an interesting principal ideal domain" (JPAA 20 (1981), 157–163).
Closely related question: Why do we care whether a PID admits some crazy Euclidean norm?
The answer is no: it is not possible, in general, to reduce a matrix over a principal ideal domain (PID) to a diagonal (or trigonal) matrix by means of elementary row and column operations. (This topic has been discussed in this MO post).
The following example is a result proved by P. M. Cohn [1, Theorem 6.1 and subsequent discussion] in 1966 and doesn't rely on some $SK_1$ obstruction.
Let $R = \mathbb{Z} [\frac{1 + \sqrt{-19}}{2}]$ and let $\theta \in R$ be a root of $X^2 - X + 5$. Then $R$ is a PID (see, e.g., Hardy and Wright) and the $1$-by-$2$ matrix $( 3 - \theta,\, 2 + \theta)$ cannot be reduced to a trigonal form $(*, \, 0)$.
Edit. The following lines aim at comparing P. M. Cohn's example with those presented by მამუკა ჯიბლაძე in his answer.
Here is a first remark which applies to both examples of PIDs. Because the Bass stable rank of $R$ is at most $2$ (use Bass Cancellation Theorem), it easily follows that any $1$-by-$n$ matrix over $R$ with $n \ge 3$ can be reduced to a matrix of the form $(*, \, 0, \dots,\,0)$ using elementary transformations. This certainly contrasts with the above example of $1$-by-$2$ matrix.
This second remark only holds for P. M. Cohn's example. Let $E_n(R)$ denote the subgroup of $SL_n(R)$ generated the elementary matrices over $R$, i.e., those matrices which differ from the identity by a single off-diagonal entry. Then we observe that $SL_n(R)/E_n(R) \simeq SK_1(R)$ for every $n \ge 3$ by the classical stability theorems [2, Corollary 11.19] and $SK_1(R) = 1$ by the Bass-Milnor-Serre Theorem [2, Theorem 11.33]. For P. M. Cohn's matrix [1, Theorem 6.1], from which we borrowed the first row, this means that $\begin{pmatrix} 3 - \theta & 2 + \theta \\ - 3 - 2\theta & 5 - 2\theta \end{pmatrix} \in E_3(R) \setminus E_2(R)$.
Eventually, in the case of the D. Grayson's PID, i.e., $R = S^{-1} \mathbb{Z}[T]$, with $S$ the multiplicative subset generated by $T$ and the polynomials $T^m - 1$, one can find $A \in SL_2(R)$ such that $A \notin E_n(R)$ for every $n \ge 2$ (I am unable to provide an explicit matrix at the moment).
[1] "On the structure of the $GL_2$ of a ring", P. M. Cohn, 1966.
[2] B. Magurn, "An algebraic introduction to K-theory", 2002.