Well... I tried it, and made things worse (risetime ~ 3 ns, fall ~ 5 ns)! After some tweaking of pullup R and a shunt capacitor on the resistor to the PNP base, the waveform at the trimpot/TD has something like 30 ns rise/50 ns fall. I'm not sure why the TD isn't "snapping" it to ~1 ns like it did when driven directly from the oscillator output.
Another problem I'd forgotten about - the Ext Trig input is a 50 ohm impedance. I added a 2N4401 emitter follower from the osc output and that drives the trigger ok.
Finally, the string of all six inverters (while driving nothing but the next one) only resulted in a 40 ns delay. That was a TI 7404. I swapped in a Nat Semi (DM7404) which was a bit slower but still 50 ns. Not enough, I need 75... perhaps a small RC between two of the inverters. Never thought I'd complain that a TTL part was too fast!
This analog/RF stuff has always confused me. As an EE I worked with microprocessors (and assembly language programming). Straight digital, it's on or it's off ;)