Could an external circuit with 74HC series IC reliably test a 24 MHz crystal?
An oscillating crystal generates a sine wave, which cannot be amplified very well by a normal digital logic inverter (especially if it has hysteresis).

A Pierce oscillator works better with an unbuffered inverter (U04 instead of 04), which has a linear operating region:


source: SN74LVC1GX04 datasheet


source: SN74LVC1GU04 datasheet

The additional inverter at the output converts the sine wave into a digital signal. (The SN74LVC1GX04 combines both inverters.)

Every microcontroller datasheet tells you that the traces between the crystal and the chip must be as short as possible; using leads of any length is likely to break the oscillation, and even the socket might be a problem.



For how to choose the components, see, e.g., Use of the CMOS Unbuffered Inverter in Oscillator Circuits.