If you put LED (with resistor) to the output, it should shine or not depending on the input (and on where the LED is connected).
If you connect the LED between output and ground, it should shine, when the output is high (and input is low) and be dark otherwise - it works ok, with both current desing and your modification. (When the upper transistor is open, current goes from VCC via upper transistor, then LED to ground, LED shines. If the upper transistor is closed, no connection to VCC, LED does not shine.)
If you put LED between output and Vcc, it should shine, when the output is low (and input is high), and be dark otherwise.
If input is LOW, the upper transistor is open, the output is HIGH, LED is between VCC and HIGH, so no current can go, LED is dark - it is OK.
No for input HIGH (and so output LOW) - the LED shoud shine.
The current schema ensure it, as on HIGH input the low transistor is open, so current goes from VCC via LED and low transistor to ground - LED shines.
In your modification HIGH input close the upper transistor, but as there is no lower transistor. So current go from VCC via LED and there is no path to ground, so no current can go and LED is dark too - it fails.