Does the volume control effect sound quality?

Big time . There's a reason high end manufacturers use stepped attenuater's .

Sowter attenuater transformer volume control. From their website:
"The transformer action of these attenuators assures the loading circuit sees a low driving impedance regardless of switch position. This ensures the maximum bandwidth potential and minimum distortion of the circuit can be realized at all gain settings."

YES! Most amp manufacturers will put in the cheapest volume pot in their amps. You'll notice the biggest improvement by swithching to a stepped resistor or transformer/autoformer volume control. LDRs will also give you a big benefit, but I don't know of a kit designed will fit into an amp.

We've done a lot of mods on Dynaco equipment over the years in fact that is how we got started back in the 70s. The volume control on the older PAS-3 was always a problem. Eventually we had some custom switches made up so we could install a stepped control in the unit.

The impedance of the control is important. The stock control on the original PAS-3 was 250K. If you install a larger value it will mess with the 12AX7 at the input of the circuit as the volume control and the input capacitance of the tube interact (Miller Effect). However if you go below 100k don't expect any bass out of the phono section!

IOW you do have to choose the value carefully. BTW it will not matter how nice the control is if these parameters are not observed- the same electrical rules apply to the best controls as the do to the cheapest.

I believe that the volume potentiometer (pot) is the single most important component that can have detrmental effect on sound quality, this is why they all sound different to each other.
All volume controls have pin point touch contact that are purposely made light in pressure as not to wear out the resistive track or contact points, all the music signal has to try to pass through these contacts.
These contacts/track are also mostly made dissimilar elements/metals which can have a "diode effect" which tries to rectify the ac music signal, this is a form of distortion.
In order what I believe is the best type of volume controls.
1: Lightspeed (ldr based) Attenuator or TVC (transformer based) = no contacts in the siganl path.
2: Mercury Wetted Relays = with gold or silver contacts in a mercury bath, they also resist tranishing /arcing, also are becoming illegal.
3: Good quality relays = with gold or silver contacts
4: Switched resistor = double leaf, each side of the wiper contact
5: Switched resistor = single leaf on one side of the wiper
6: Potentiometer (Pot) = conductive plastic track
7: Potentiometer (Pot) = carbon track

Cheers George