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At TTL voltages.That's odd that it can source more than it can sink.
At TTL voltages.
It can source a higher current when its voltage drop is allowed to be half the supply voltage.
It can sink a small current when its voltage drop is only 0.4V.
The datasheets from Texas Instruments have graphs that show the output current at three supply voltages and with any amount of voltage drop. The current is pretty high (typically 14mA) directly into an LED when the supply is 10V.
Then the typical output current directly into a red LED is 3.5mA.The OP runs his circuit at VDD=5V.
Then the typical output current directly into a red LED is 3.5mA.
I buffered the output through a 4013, it's still not working.