.... I would prefer a PC software solution though. Why doesn't this exist?
How would you connect all those serves to a PC? Better to use some kind of micro processor
How do people program servo sequences? Just code in line-by-line position commands to the servos?
For each servo you write an equation that give position as the function of time.
Then you write a loop that runs say 30 times per second. It could run off a timer or just poll the system clock.
Either way the loop runs at 30Hz (or 5Hz or 100Hz or whatever you like)
Then inside the loop, for each servo, the software gets the time and uses your equation to convert time to position and sends that to the servo.
The trick is writing those functions. In a typical robot you'd need to use an inverse kinematic solution. What this does is convert (x,y,z)) space motion to servo angles. Say you want to move the arm form (3,6,7) to (4,9,12) in one second and the arm as five servo motors. 30 times per second you figure out intermediate X,Y,Z coordinate and then the servo angles and then write to the servo to angle. The arm then moves in a straight linear line.
Likely it is more complex because you want the arm to have zero speed at the start and end of the movement and some maximum speed at the mid point you you don't linearly interpolate step on the line (3,6,7) (4,9,12)
One more thing, before you do the move you'd want to check that you are not driving any motor after than it can go
Bottom line: For smooth motion, your software needs to update the position of each motor many times per second. If the motors are serially connected to make a robot arm the math gets a little complex and forces you to remember your trig and geometry.