/bin/bash not executing any commands
I have found the culprit:

if [ -f $(brew --prefix)/etc/bash_completion ]; then
  source $(brew --prefix)/etc/bash_completion
fi


These lines in my ~/.bashrc were causing bash to hang up on startup and it wasn't executing any commands as a result. Commenting them out solved the issue.

Apparently, there is some error in that script (it's part of Homebrew's bash-completion package).