Why doesn't using multiple commands with a || or && conditional work?

Note that your line

[ "${#}" -eq 1 ] || echo "Invalid number of arguments, expected one."; exit 1

this is the same as

[ "${#}" -eq 1 ] || echo "Invalid number of arguments, expected one."
exit 1

(an unquoted ; can, in most circumstances, be replaced by a newline character)

This means that the exit 1 statement is always executed regardless of how many arguments were passed to the script. This in turn means that the message The given argument is empty. would never have a chance of getting printed.

To execute more than a single statement after a test using the "short-circuit syntax", group the statements in { ...; }. The alternative is to use a proper if statement (which, IMHO, looks cleaner in a script):

if [ "$#" -ne 1 ]; then
    echo 'Invalid number of arguments, expected one.' >&2
    exit 1
fi

You have the same issue with your second test.

---

Regarding

[ -z "" ] && echo A || echo B

This would work for the given example, but the generic

some-test && command1 || command2

would not be the same as

if some-test; then
    command1
else
    command2
fi

Instead, it is more like

if ! { some-test && command1; }; then
    command2
fi

or

if some-test && command1; then
    :
else
    command2
fi

That is, if either the test or the first command fails, the second command executes, which means it has the potential to execute all three involved statements.

This:

[ "${#}" -eq 1 ] || echo "Invalid number of arguments, expected one."; exit 1

is not:

[ "${#}" -eq 1 ] || { echo "Invalid number of arguments, expected one."; exit 1; }

But instead is:

{ [ "${#}" -eq 1 ] || echo "Invalid number of arguments, expected one."; } 
exit 1

Your script is exiting regardless of how many arguments you passed to it.

One way to make it more readable is to define a die function (à la perl) like:

die() {
  printf >&2 '%s\n' "$@"
  exit 1
}

# then:

[ "$#" -eq 1 ] || die "Expected one argument, got $#"

[ -n "$1" ] || die "Empty argument not supported"

You can add more bells and whistles like colours, prefix, line number... if need be.