The point is that the order in which that is processed is not left to right.

First the | pipe is established as fd [1]. And then 2>&1 duplicates that pipe into [2]. I.e. right to left: opposite to left-to-right processing of redirections.

When you need to capture both standard error and standard output to a file, you must have them in this order:

  bob > file 2>&1

  bob 2>&1 > file

But if you change > file to | process, then it's fine! process gets the combined error and regular output.