8 servo's, each of which need to perform a function on a loop with a specific delay in between. So: action - delay in ms - action - delay in ms etc...
The thing is, the 8 servo's need to do this independently from one another (each servo would have the same action, but with different delays in between the action).
The delay of each servo also needs to be variable and controllable by a rotary pot or rotary encoder. This could mean 8 rotary's, or just one rotary with a switch to change between the servo's. A readout of the delay in ms would be nice too.
Any advice on which Arduino board would be best suited for this?
The first help any one will supply is: Any Arduino will do your project. How long have you been programming with C? If not long, start your learning by forgetting the project and begin to exercise EACH of the sample programs that come with the Arduino IDE system that you have installed on your computer.
Eventually you will have to decide how much torque is required from each of your servos before you know what to purchase.
Good luck with a very rewarding hobby/education.
if you use a rotary encoder which incorporates a switch and a LCD you select a servo by pressing the switch until you get the required servo as shown on LCD and then rotate to the required setting as shown on the LCD
any other requirements? WiFi, bluetooth, ?????
My recommendation is to use an Arduino Nano. This Arduino offers eight analogue inputs if you want to use the mentioned potentiometers.
My second recommendation is to use OOP to design the sketch.
The programme contains eight configurations for the servo/pot hardware and a timer function to control the timing of the hardware.
almost any microcontroller would meet your requirements
I would tend to go for an ESP32 as the experience of using such a modern microcontroller would probably be useful for future more sophisticated projects
You will not find silent servos, but you can make boxes around the servos that absorb the sound.
Take a look at this example simulation of multi-servos with multi-controls, showing the "non-blocking" thing. The simulation does not include all the input devices (analog knobs) that you want, so you will need to find the right multiplexer for your purpose.
And here is some further reading about PWM and using a logic analyzer on a simulation: