Without even knowing the detail of the potentiometer control of the
Huanyang HY01D523B VFD, I can share some of what I know about most all of these I have encountered. Most have an internal control way of altering the voltage, sometimes by pot, sometimes all digital, but for those which have a rotation pot, the external control ends up in effect simply wired in parallel. There is the reference voltage across the pot. It might be +5V. Some are 10V, some are between +5V and -5V.
The actual value of the reference voltage does not affect the proportional division of the pot wiper voltage. It can be anything the manufacturer designs in. The track of the pot is linear. It is not like the logarithmic variation as used for audio electronics volume control. The proportion of the the speed setting is a signal input speed demand to the VFD control.
The internal circuit might have have some ballast resistors either side of the control pot, to set the highest and lowest voltages the design might require, as well as safety limit the current in case of pot short or miss-connection. My first thought corresponds with yours, that is, to try a lower impedance (ie resistance) control pot. Say 5K, or 1.5K. Go carefully! On many systems, the external control has to be lower impedance that the internal control, so that it runs perhaps 5x or 10x the through current, to have enough authority to override the internal control.
The connections have to be correct!
The most common reason wired up controls deliver only partial control range is because one has wired the pot with the wiper swapped for one of the ends. This can be very bad, because at one end or the other, one could be short-circuiting the reference voltage. Good designs limit the current to the control.
Connecting these things up does not require much electronics knowledge, but one can make mistakes. There is no substitute for researching what is the correct control resistance. Very important is to get the connections right!
Your max speed arrives early
That you get the max speed when the control is only 15% of the rotation span, I think, is not totally fixed by changing the value of the pot to 1.5K, although the correct recipe is as you suggest. Strictly, you have to account for the 85% of the resistance that was still in there (as further to go), when you hit the maximum. Adding some ballast resistance to the higher voltage side of the pot allows more of the physical rotation span to represent the speed demand range.
This symptom still sounds like it could be wiper swap for one end miss-connection, but I am assuming you would get that right.