There are a lot of questions rolled into one here, and I can't answer specifically to the F-16, but a turbine engine needs the same three elements of the basic fire triangle to run: Fuel, Air, Ignition. The element missing from the sequence description linked in the question is ignition.
The igniters are like spark plugs, and they are probably energized during actuation of one of the other mentioned switches. Most likely the JFS switch has a secondary function of energizing the ignitors since it is engine start specific.
Airstarts are not an overly complex procedure, in fact for a single engine aircraft like the F-16 they need to be brief and committed to memory because fighters don't have the best glide ratio.
Fuel - Make sure the throttle wasn't inadvertently shut off, likely the checklist will call for placing this in the idle position. (you need fuel flow, but don't want to dump too much gas into an engine that isn't lit yet!) Perhaps there is a step to turn on a boost pump or switch tanks.
Air - This comes from outside, as measured by airspeed. The minimum airspeed required envelope will vary with altitude, but most pilots will have a number committed to memory that will provide reliable airstarts across a broad range of altitudes.
Ignition - There will likely be a separate igniter switch for airstarts. Press and hold until lightoff is achieved, then advance the throttles smoothly and resume flying. The airstart ignitor switch is generally located on the power control lever for convenience. If single engine, an alternate electrical power source may be required first in the form or activating a Ram Air Turbine or other source of emergency power to provide a spark if the generator is disabled.
The process isn't really any different from lighting a natural gas stove or propane grill that has a piezoelectric lighter - You turn on the gas and hold the button down or press repeatedly to make the igniter spark until the flame lights, then release the button. (in fact when testing the airstart igniters on the ground before flight you listen for a clicking sound to indicate they are sparking!)
So, in its simplest form a generic airstart checklist for a tactical jet would something like the following example:
- AIRSPEED - Maintain 200 KIAS minimum.
- THROTTLE - Idle.
- AIRSTART BUTTON - Press and hold until light off.
- If no relight by 5000 feet AGL - EJECT