Accuracy
Trefethen and Schreiber wrote an excellent paper, Average-case Stability of Gaussian Elimination, which discusses the accuracy side of your question. Here are a few of its conclusions:
"For QR factorization with or without column pivoting, the average maximal element of the residual matrix is $O(n^{1/2})$, whereas for Gaussian elimination it is $O(n)$. This comparison reveals that Gaussian elimination is mildly unstable, but the instability would only be detectable for very large matrix problems solved in low precision. For most practical problems, Gaussian elimination is highly stable on average." (Emphasis mine)
"After the first few steps of Gaussian elimination, the remaining matrix elements are approximately normally distributed, regardless of whether they started out that way."
There is much more to the paper that I can't capture here, including the discussion of the worst-case matrix you mentioned, so I strongly recommend that you read it.
Performance
For square real matrices, LU with partial pivoting requires roughly $2/3 n^3$ flops, whereas Householder-based QR requires roughly $4/3 n^3$ flops. Thus, for reasonably large square matrices, QR factorization will only be about twice as expensive as LU factorization.
For $m \times n$ matrices, where $m \ge n$, LU with partial pivoting requires $mn^2 - n^3/3$ flops, versus QR's $2mn^2 - 2n^3/3$ (which is still twice that of LU factorization). However, it is surprisingly common for applications to produce very tall skinny matrices ($m \gg n$), and Demmel et al. have a nice paper, Communication-avoiding parallel and sequential QR factorization, which (in section 4) discusses a clever algorithm which only requires $\log p$ messages to be sent when $p$ processors are used, versus the $n \log p$ messages of traditional approaches. The expense is that $O(n^3 \log p)$ extra flops are performed, but for very small $n$ this is often preferred to the latency cost of sending more messages (at least when only a single QR factorization needs to be performed).