Incorrect frequency from crystal oscillator
With the details provided, it does seem like the crystal may be damaged - it's the most sensitive component on the board, and there's a good chance the pulser has been dropped a few times during it's long life. 

The circuit seems reasonable and if it was originally designed with the resistor (rather than the capacitor), then I'd leave it like that. There are similar circuits out there that do use a capacitor as you say:



I'd go for replacing the crystal - choose a crystal designed for series operation such as the FOXLF160 (as opposed to the FOXLF160-20) While you're grabbing the crystal, getting a couple of 74LS38s wouldn't be a bad idea either, given they are only around 50 pence/cents each.  

Also, given the price of decent CMOS oscillators nowadays, the other option is just to buy a 16MHz CMOS oscillator (5V supply tolerant) and replace the first stage with that.