When you added the photo, I realised I had a couple of them in my 'spares' box, which I had added to an AliExpress order, to look at on a rainy day.

As the weather looked like it might rain, I rigged up a simple test with 6V battery and multimeter, and found the same as you did. They seem to be made with a weird potentiometer that gives a relatively wide mid-section, with almost no change; then a narrow section either side of the mid-section, which goes from 50% to 0%, or 50% to 100%, and outside of those sections, a pair of wider sections, which are either all 100% or all 0%. So the only parts with a graduated change, are two very small angle sectors, of maybe 10 degrees each.

So, in spite of looking to be 'fairly' well-made, their utility seems limited. However, I only paid about US $1 for a pair (including delivery), so I don't have too much to complain about.

I realise that you found the built-in switch convenient, but I notice Ali-Express also do larger joysticks, which superficially look like the type I have seen on radio control transmitters, etc. They are a variety of (higher) prices, but might be better if you wanted control in 2 dimensions, but maybe not including a switch. Sorry, I have no experience of them, nor do I have any in the 'spares' box.

Good luck with your quest. Dave

Sorry but I can't make a recommendation. I don't ever recall buying a joystick.

I'm thinking of trying a different type of control using a slide potentiometer. 

I think I have a few of those sticks, so if my heart is up to it tomorrow I will test all of them and report back.

In this case, the values change so quickly that it makes it difficult to do a slow roll up on the motor to maximum speed.  I want to use this type of joystick because of the built in switch that allows me to run two motor independently. 

I think I paid $5 each.  Would you be able to suggest a better one?

https://amz.run/9IhX

  Sorry, I don't have any experience of commercial joysticks, but as @will has mentioned, they usually use a potentiometer, and any of the mechanical defects mentioned sound feasible.

However, if it were mine, as a kind of cross-check, I would start with using a multimeter to measure the voltage that the joystick is sending to the processor changes smoothly over the whole range of movement of the stick, and that all of those voltages are within the range of the analogue-to-digital converter.

Remember, analogRead, does not 'actually' measure resistance, but voltage, so this is using the meter to directly display the voltage that analogRead is 'seeing'.

In particular, if the processor is a 3.3V, make sure the energising voltage provided to the joystick is not 5V. 

https://www.arduino.cc/reference/en/language/functions/analog-io/analogread/ lists the Arduino processors that are 3.3V, with other boards like Raspberry Pi Pico, ESP32, ESP8266, also being 3.3V processors.

Note, this is only a 'last chance saloon' check, but it might give you a little more insight into what is happening, and how to fix it.

Best wishes, Dave

It this due to the quality of the joystick?

Probably, but it may be poor construction of the joystick itself (was it a cheap product ?).

There are several alternatives, all unpleasant. You can a) try a higher quality (cost) joystick; b) glue a barrier onto your existing joystick so that the travel and potentiometer match, c) make a do-it-yourself joystick with a potentiometer, d) try taking the joystick apart and carefully repositioning the built-in pot. That last method might work if the stick seems to give a reading before it's moved and stops changing before it finishes. Such behaviour indicates that the pot is mis-aligned.

Using analogRead doesn't matter, it's just reading the effective resistance of the pot and has absolutely nothing to do with the relative positioning of the joystick lever. So its not causing a problem and can't be used to correct the joystick.