As an aside: a solid (no moving parts) but radiused tappet has been used for many decades in English motorcycles (Triumph, BSA, Norton). This allows more choices of lobe shape, acceleration etc. as described for roller tappets, but of course still has sliding friction, break-in problems, life expectancy etc. This was dealt with by use of very light components, low ratio rocker arms (which reduces the spring load on the tappet) and frequent oil changes!
Triumph also used at least 2 radii for the contact surface: “Std.” with ¾” radius and “R” with 1â…›” radius. As described, an “R” tappet will produce slightly more duration than a “Std.” tappet on the same lobe. These could be interchanged to make minor tweaks in the engine's "tune up" without replacing or re-timing the cam.
Harley-Davidson used an asymmetrical mushroom tappet from the 1941-52 in the 45" race engines. This had a .731" stem (body diameter) but the "foot" was over 1" in length.
Obviously, any non-rotating tappet needs an alignment fixture.
Yes, they both developed a flat spot in the contact surface half-way along where the lobe peak landed.