Pull-up and Pull-downs are normally used to ensure a line has a defined state while not actively driven. They are used on inputs to prevent floating lines, rapidly switching between high and low and a middle "undefined" region. Outputs normally do not need them.
But most mcu pins are GPIO, and sometimes on startup are defined as inputs instead of outputs. As you said, sometimes you don't want an IC pin input floating on startup, especially like a reset pin that you would normally drive with your microcontroller's GPIO.
This is when you use a Weak Pull-up or Pull-down on the line. Because they are weak, and you choose the default state, they provide no interference with your circuit (If the input is supposed to be low at all times, then pulled high, you choose a weak pull-down, and vis versa), but they do consume a bit of current. This is why you choose a resistor weak (Higher the value, the weaker) enough for the job.
Another normal output setup that uses pull-ups (or pull-downs, rarer) is Open Drain or Open Collector connections. These only drive a connection low, or release the line, leaving it floating. The pull-ups are used to bring the line to a high logic state.
An open-collector or open-drain output would always need a pull-up resistor to get a defined voltage level. But you are not using such an output.
TTL outputs have defined voltage levels for both low and high signals, but the high level can be as low as 2.4 V (depending on how much current the output must supply). This is no problem for TTL inputs (which interpret any voltage above 2 V as high), but can be a problem for other logic families.
CMOS outputs go up to almost the supply voltage, and go down to almost ground level (because they do no use bipolar transistors but MOSFETs). Therefore, CMOS inputs are quite strict (typically, low-level signals must be below 30 % of VCC, and high-level signals, above 70 %).
If you want to connect a TTL output to a CMOS input, you need a pull-up resistor to raise the high level signal. (Connecting a CMOS output to a TTL input works just fine.)
(In the datasheets, the guaranteed output levels are specified as VOL and VOH, and the required input levels as VIL and VIH. An output and an input match if \$V_{OL} \leq V_{IL}\$ and \$V_{OH} \geq V_{IH}\$.)
LS is a TTL family; HCT is a CMOS family that has TTL-compatible inputs. So in your case, you do not need a pull-up resistor to get a correct voltage level.
There might be other reasons to use pull-up/-down resistors, for example, to get a defined signal when the chips are still in reset and do not drive their outputs either way.
Best Answer
The pull-up or pull-down would make sense if the IO pin is configured to be an input, or at least a pull-up would make sense if it is configured to be an open-drain output.
Not all combinations that are possible to select in hardware and software are useful. It really does not make sense to be a push-pull output with pull-down resistor enabled.
But, for example, it might make sense to have a pull-up or pull-down, if the IO pin is for example used as a bi-directional data bus for a memory chip, or SDIO card, or something along that line.