74hc595 sinking, not sourcing
It looks like we’ve encountered some resistance.



The Pch and Nch have a wide tolerance for Vt near 1.25V and if the Vt of Pch is higher than the Nch this would explain the inability to drive the LED.

Although rated for 2~6V for the 74HC' CMOS family, the worst case specs for (Vdd-Voh) match Vol so it is the symmetrical worst case but typically, one may expect slightly weaker Pch or higher Vt thresholds.

However the guaranteed IC std. test currents are as follows;
at 2V @ 25'C,  Vol/Iol = 0.1V/20uA = 5k Ω = Ron " RdsOn "
at 4.5V @ 25'C,  Vol/Iol = 0.26V/4mA = 65 Ω
at 6.0V  @ 25'C,  Vol/Iol = 0.26V/7.8mA = 33 Ω
Thus interpolating to 3.3V I expect RdsOn may 400~500 Ohms  is wishful thinking.

Conclusion:

For your prototype, you can bypass the 1K resistor.

For reliable LED driving 3.3V RdsOn performance, you can switch to the 74VHC' CMOS family 74VHC595 which is 360mV/4mA= 90 Ohms @ Vdd=3.0V