I read your post and thought I should reply.
About 10 years ago I worked for a Chrysler dealership. One of the problems some motors with aluminum block encountered was a thing called piston slap.
This would occur at cold start up then quiet down once warm.
My understanding is that until the rings warmed and expanded, the piston was slapping within the cylinder.
This piston slap sounded more like a knock knock than the click click of a tappet.
A bent rod can increase the contact distance and cause tappet noise.
A motor which has been run low on oil and ran hot can develop noisy tappets among a list of other things.

I'm no mechanic...