TTL crystal oscillators are vanishingly rare nowadays and much of the common knowledge about them has been largely lost.
What you've got there is a variant of the capacitor coupled two inverter series resonant oscillator circuit as the low input impedance and relatively low gain of TTL inverters made Pierce oscillators rather difficult to implement reliably.
I'm not at all surprised that it's sensitive to the brand of inverter used as using a TTL inverter as a linear amplifier isn't specified in their datasheets, so yes even the tiny differences you have noted are important. Although the circuit could be improved to ensure reliable operation with most brands of 74LS04, as long as you can 'select on test' to find working ICs for it, loosing originality by modifying vintage computers is generally undesirable. Maybe build a test jiq to aid selecting inverters that work?
You should read Chapter 11 of "Crystal Oscillator Circuits" by Robert J. Matthys, Revised edition: Krieger Publishing Company, 1992, ISBN 0894645528. It was originally published in 1983 by Wiley, and is out of print in both editions, but PDFs of it are easily Googlable.
Here are a few select quotes:
In general, TTL does not work too well in an oscillator circuit, because its input resistance is highly nonlinear at the switching point, and it exhibits bad parasitic oscillations if it does not switch rapidly between its two binary output states.
11.3. TTL TWO-INVERTERS-7404
... Many versions of this circuit are poorly designed; ...
It then goes on to explain the correct biasing and operation of this type of dual inverter series resonant oscillator in some detail, for different series of TTL logic. See: Figure 11.3(c). TTL Two-Inverters (74LS04) circuit, which has very different biasing to your circuit.