I need a special LED chase circuit. The chase is initialized by a physical event with a momentary
switch closure.

With the switch closure, the LEDs will chase with each LED remaining on until the chase completes with a
maybe .5 sec delay before turning off.

The chase speed needs to be adjustable. The ON delay can be fixed, but adjustable through component
changes.

 

A mcu (like an arduino) would do it with ease.

 

Looks like you have it, just convert block to schematic. Also the bar graph can be constructed with the 4017 feeding OR gates rather than the novel use of SCR's.