Those springs are called "dampener springs" , and began to be used in clutch discs in the early 1930's.
By about 1935, the only company using a non-dampened disc was Hudson, and that was really a different animal, being a cork-faced "wet-clutch".
The purpose of the damper springs in the clutch hub is to promote smoother engagement.
Old-time solid-hub clutches can tend to "grab" or chatter on engagement, particularly if the driver has the engine speed "up" a bit, and is trying to gently "slip the clutch" to get the vehicle moving; it will tend to do a series of "bunny-hops"...
The "modern" clutch disc has two spring elements to it: the four to six damper springs around the hub, and the flexible springy metal core of the disc.
Modern clutches will probably engage more smoothly; service life will depend on the lining material used and the habits of the driver... in general, the less time the clutch spends "slipping", the longer the linings last.
When I'm driving any stick-shift vehicle, I try as much as possible to synchronize engine & driveshaft speed to minimize slipping the clutch.
For getting the vehicle under way, I will usually goose the throttle, and as the engine is coming back to idle, I am letting the clutch out and letting the inertia of the engine & flywheel get the vehicle moving, then when I've felt the clutch "hook-up" (positively engage), I step on the accelerator and go. This is easier with old torquey long-stroke engines then it is with modern short-stroke cars...
Aside from changes in friction material, there really haven't been many(any?) improvements in clutch design since the "cushion-plate / damper-hub" clutches of 1935, such as Bruce pictured in reply #2...
I cannot think of a reason NOT to use the "modern" type clutch disc in a Model A, unless you are going for an absolutely authentic restoration.