List all binaries from $PATH

This is not an answer, but it's showing binary, a command which you could run

compgen -c

(assuming bash)

Other useful commands

compgen -a # will list all the aliases you could run.
compgen -b # will list all the built-ins you could run.
compgen -k # will list all the keywords you could run.
compgen -A function # will list all the functions you could run.
compgen -A function -abck # will list all the above in one go.

With zsh:

whence -pm '*'

Or:

print -rC1 -- $commands

(note that for commands that appear in more than one component of $PATH, they will list only the first one).

If you want the commands without the full paths, and sorted for good measure:

print -rC1 -- ${(ko)commands}

(that is, get the keys of that associative array instead of the values).

In any POSIX shell, without using any external command (assuming printf is built in, if not fall back to echo) except for the final sorting, and assuming that no executable name contains a newline:

{ set -f; IFS=:; for d in $PATH; do set +f; [ -n "$d" ] || d=.; for f in "$d"/.[!.]* "$d"/..?* "$d"/*; do [ -f "$f" ] && [ -x "$f" ] && printf '%s\n' "${x##*/}"; done; done; } | sort

If you have no empty component in $PATH (use . instead) nor components beginning with -, nor wildcard characters \[?* in either PATH components or executable names, and no executables beginning with ., you can simplify this to:

{ IFS=:; for d in $PATH; do for f in $d/*; do [ -f $f ] && [ -x $f ] && echo ${x##*/}; done; done; } | sort

Using POSIX find and sed:

{ IFS=:; set -f; find -H $PATH -prune -type f -perm -100 -print; } | sed 's!.*/!!' | sort

If you're willing to list the rare non-executable file or non-regular file in the path, there's a much simpler way:

{ IFS=:; ls -H $PATH; } | sort

This skips dot files; if you need them, add the -A flag to ls if yours has it, or if you want to stick to POSIX: ls -aH $PATH | grep -Fxv -e . -e ..

There are simpler solutions in bash and in zsh.