I'm not convinced that the clock will oscillate.

AIUI, the idea with CMOS oscillators is that you bias an inverter into its "linear" region with a feedback element such as a resistor. The crystal + capacitors then "nudges" the circuit to oscillate at the desired clock frequency. At one point Intel had an excellent App note explaining this, but as I can't locate it, here is one from Maxim:
https://www.maximintegrated.com/en/app-notes/index.mvp/id/21541) It looks like the feedback from the output pin is missing.
<pedant on> 74HC isn't TTL, it is CMOS. TTL is plain old 74/74LS etc. 74HCT is also CMOS, but with I/O levels better suited to interfacing with "old" TTL<pedant off>
2) Diode gates work, but require one diode per input, plus one resistor. You need to choose a suitable output resistor for the speed of input/output, based on the capacitive loading of the output. You also lose 0.3-0.6V of noise immunity. For example say the input to a diode "or" is 2.9V, the output might only pull up to 2.3-2.6V (as the diode will have a voltage loss) - meaning that the next input is more prone to responding to "glitches". Given that you can get single gates in ICs these days, I'm not sure if diode gates are as useful as once they were.
3) TTL is relatively power hungry (although 74LS is less so than 74 for example), CMOS isn't (except at high speeds - 32KHz isn't). The power hungry items are the displays. For example, I have a fancy wall clock with 12 x 7-segment displays for time etc. It draws about 20mA with no display, and 120mA when the displays are lit
If you haven't already, I would suggest reading a copy of "The Art of Electronics".