different value than 500Kohm

The input stage is taken from the MXR microamp, you can have a look at it:
www.electrosmash.com/mxr-microamp

If gives an input gain from 1 to 20 (which is very versatile for electric guitars), if you have a look to the schematic:
www.electrosmash.com/images/tech/pedalsh...ut-output-stages.jpg

The gain formula is calculated as:
Gv = (1 + (R5/Rv1+R4))
In the original circuit we have:
Gvmin = (1 + (100K/500K + 4.7K)) = 1
Gvmax = (1+(100K/0+4.7K)) = 22

If you want to use a 10K pot instead of the original 500K, you would need to change some other values:
Gvmin = (1 + (4.7K/10K + 200R)) = 1.4
Gvmax = (1 + (4.7K/0 + 200R))= 24

side effect:
To keep the filters where they are you would also need to mod the cap C4:
The original C4 and R5 keep a low pass filter of around 5KHz, to keep a similar freq you would need to change C4 for 6.8nF
This filter is important to reduce the background noise.

In a nutsell, to change the 500K to 1K you would need to also change other components:
originals:
R4=4.7K
R5=100K
RV=500K
C4=270pF

your change:
R4=200R
R5=4.7K
RV=10K
C4= 6.8nF

note: If you have a 100K pot (instead of 10K), you can keep the same components the same.

I notice that even with 500k linear most the the gain effect seems to be toward one end of the pot, rather than evenly spread.

yes, I also noticed that the pot action is not evenly spread.

We could fix that using a log ( or anti-log ) resistor trimmer instead, but these parts are not as easy to find and cheap as simple linear trimmers. This part is supposed not to be adjusted all the time, so we decided to use a common part sacrificing a bit the user experience.