Tuning BJT avalanche noise
With Q2's collector resistor of 10k, more than about 11 mA collector current will cause Q2 to go into voltage saturation - you want less than 11 mA peak collector current. It is not easy to guess what base current is required, since Q2's current gain can vary over a wide range (perhaps 100 to 300).

And the reverse breakdown voltage of Q1's base-emitter junction is not known either. It is usually around 7-8 V (data sheet says it could be as low as 6 V). Q1's base-emitter junction has to reach this voltage to generate even a little current for Q2's base. Transistors other than 2N3904 are built differently, and may have much different breakdown voltage \$ V_{(BR)EBO} \$. 
You'd like Q2's base current to be somewhere in the 16 - 50 uA ballpark, depending on Q2's current gain. This current is generated by Q1 only when voltage across it exceeds its breakdown voltage. So you're quite right...you only have to exceed breakdown voltage a little bit to generate noise current. But too much noise current saturates Q2: the "sweet-spot" window is narrow.
The Thevenin equivalent resistance \$ R_T \$ of the variable resistor might be about 1000 ohms. If \$ V_{(BR)EBO} \$ is 7 V, then any Thevenin voltage \$ V_T \$ above about 7.1 V will saturate Q2 with too much base current.