AWOL:
I don't want it to wait (even a few milliseconds) for a servo to position
It's a mechanical device; if it has to move, it may take tens or hundreds of milliseconds to get to where it is told.
If you want faster, you need a bigger energy budget.
He means he doesn't want the program to pause while it moves into position.
Now, for answering the question.
The trick here is to prioritise. First make a list of everything that is going on.
Then, split that list into two sub lists - a list of things that you need to happen "NOW!!!", like reacting to encoders, etc, and one of things that can take time to happen, and you don't need to know about them immediately.
The things you need to happen "NOW!!!" you want to use interrupts for. Everything else you just have running in your main loop() and use polling for inputs, etc.
Then you implement a Finite State Machine. Or, probably more accurately, a number of small Finite State Machines.
Each component, or sub-assembly can have its own FSM, such as a simple little one for a servo, which has the states "IDLE", "MOVING", "ARRIVED", for example. Each FSM will have its own set of variables, for storing the current state, and any parameters for what it is operating.
The interrupts that get triggered when important events occur can do small amounts of processing of the incoming information or stimulus, and if needed modify the state or parameters of one or more of the FSMs.
Edit: I have grown so tired of explaining FSMs that I have written a tutorial all about them: http://hacking.majenko.co.uk/finite-state-machine