Yep, usually the "click" heard is the sound of the tips of the trigger's forward prongs (or "ears"), as they're pulled down back into the V-notch on the back of the drawbar head, after having slipped up above the top edge of the V-notch at that spot.
Sometimes the spring tension against the back of the trigger's prongs may be reduced enough to allow the trigger prongs to tip upward at the front and slip above the top edge of the V-notch. Pulling the trigger brings them back down into the notch ... but produces that clicking sound.
I've been told by different factory folks throughout the years that while it's possible to have the SA trigger click occur in other calibers of the 3rd gen guns, it's most likely to occur in the .45/10 guns. Something about the dimensions and tolerances involved in the frame & drawbars in the .45/10's.
I have found it occurring in an occasional double column 59XX 9mm, though.
Anyway, while a slight adjustment of the trigger play spring can usually resolve this minor "click", it may not be a permanent fix.
The older style trigger pay spring, with the separate leaf springs on each side ("open" at the top of the spring), were somewhat stiffer than the current spring.
The current spring is connected across the top, and the side leafs are a bit less stiff. I've seen how if one side of an older version of the trigger play spring were to snap off, the remaining side could still be tensioning the spring (but I replaced the broken spring, anyway).
I'm not privy to why the engineers made the spring change from the separated sides to the connected sides, from thick/stiff to thinner/lighter, but the new style seems less likely to snap off, being more flexible and "forgiving".
Of course, excessively "adjusting" either style can accelerate the normal wear & tear on the spring, and even break them. Some caution would seem prudent in being over zealous in frequently "adjusting" the springs.
Anyway, the guns will run without them. The SA trigger will just have a bit more "slop" (forward & rearward), which probably wouldn't be noticed during the stresses of actual shooting ... unless someone was doing slow-fire target style shooting in SA. I remember firing approx 800+ rounds during 2 days of training & practice some years back, with a 6906, during which time the trigger play spring completely snapped off my gun. I was surprised to see it missing, as I hadn't felt anything different at all throughout my shooting.
Some years ago the CHP out here in CA apparently tired of shipping guns back to their Sacto gunsmiths to have the occasional broken trigger play spring removed, as I was told they removed the springs from all their guns and even drilled out the drawbars so a field armorer couldn't replace them (I was told this by one of their field armorers). When they ordered their new 4006TSW's, this time around they were ordered produced without trigger play springs.