The order in which init files are sourced is wrong for a interactive login shell. Apparently because of https://bugs.archlinux.org/task/15456.

Current order:

/etc/profile (sourced by bash)
/etc/bash.bashrc (sourced by /etc/profile)
/usr/share/bash-completion/bash_completion (sourced by /etc/bash.bashrc)
~/.bash_profile (sourced by bash)
~/.bashrc (sourced by ~/.bash_profile)

But /etc/bash.bashrc should be sourced in ~/.bash_profile instead (which makes all the tests of /etc/profile obsolete at the same time).
For example /usr/share/bash-completion/bash_completion (sourced in /etc/bash.bashrc) is sourced before user defined variables are set in ~/.profile, which prevents the use of the BASH_COMPLETION_USER_FILE variable (sourced at the end of /usr/share/bash-completion/bash_completion).
Correct order in which profile files are sourced before anything else:

/etc/profile (sourced by bash)
~/.bash_profile (sourced by bash)
/etc/bash.bashrc (sourced by ~/.bash_profile)
/usr/share/bash-completion/bash_completion (sourced by /etc/bash.bashrc)
~/.bashrc (sourced by ~/.bash_profile)

Other solutions are:
- defining BASH_COMPLETION_USER_FILE in /etc/profile (impossible for non-root user)
- sourcing /usr/share/bash-completion/bash_completion in ~/.bashrc (so after ~/.bash_profile) instead of /etc/bash.bashrc

What need to be removed in /etc/profile:

# Source global bash config, when interactive but not posix or sh mode
if test "$BASH" &&\
test "$PS1" &&\
test -z "$POSIXLY_CORRECT" &&\
test "${0#-}" != sh &&\
test -r /etc/bash.bashrc
then
. /etc/bash.bashrc
fi

What need to be added at the top of /etc/skel/.bash_profile (and every existing user's ~/.bash_profile):

[[ -f /etc/bash.bashrc ]] && . /etc/bash.bashrc

New users will have nothing to do, but existing users will have to add the /etc/bash.bashrc sourcing in their ~/.bash_profile.