I only looked at it briefly but it seems to me that the idea is that the output of the LS30 will be low if no button is pressed and high if a button is pressed. this in turn is inverted by the LS00 which then is input to /LE on the LS373 and thus latching up the pressed button(s). so, I would follow those signals - pin 11 of the LS373 should be high when no buttons are pressed and then low when one has been pressed. If that's not the case, then trace the signal back. Based on your description, /LE is always high - I'd look at the LS30 and LS00 to see if they are properly connected.
Now, reset is a bit odd to me. pin 1 of the LS373 is output enable. pulled low, it enables output. pulled high, it puts the output pins in high impedence. I can only surmise that this is supposed to cause the LS30 to see high outputs from the LS373 and output a high which causes the state of the buttons to pass through the LS373. seems wrong to me. but presuming it can actually work that way, check that you have pins 1 and 11 of the LS373 properly wired - your schematic is unclear on that point.
Maybe a better solution for a reset is to use s1 to pull /LE high. this requires that no one is pushing any buttons. The LS373 will pass though the button signals (which are high), thus causing the LS30 to assert /LE high when s1 returns to the unpressed state. I'd use a resistor to ensure that you don't get a short circuit, though. debounce is a good idea but not really necessary, assuming that no one touches the buttons.