So if assemble the exact circuit as shown and I use a trimmer instead of common potentiometer, would I have a reasonable range of frequency?
This is something you can easily predict. You don't even need a calculator, or a pencil and paper; you can do it in your head.
Imagine you have a 100 kΩ, 10-turn trimmer for VR1, connected so that at full counterclockwise rotation you have maximum resistance and therefore minimum frequency. Call that frequency Fmin. At full clockwise rotation you have maximum frequency (whatever that turns out to be; for this circuit, it's undefined).
Starting at full counterclockwise, and Fmin, you have to rotate the trimmer 9 full turns clockwise to get it down to 1/10th its full resistance, and therefore obtain a frequency of 10 * Fmin. The next 0.9 turn clockwise gets you to 100 * Fmin, and the next 0.09 turn after that gets you to 1000 * Fmin, and so forth.
As you can see, as you go higher and higher in frequency it becomes more and more difficult to set a precise frequency, to a point where moving the trimmer just the tiniest amount results in a drastic, uncontrollable frequency change.
So the practicality of this scheme is going to depend partly on what you consider a "reasonable" frequency range, and partly on how much annoyance you're willing to endure when working with high frequencies and attempting to set them precisely.