In Newtonian Mechanics, the rest frame of a lab on the Earth is considered to be (approximately) an inertial frame. The fact that a thrown ball is not moving on a straight line corresponds to the action of the force of gravity.
In special relativity, we don't have a description of gravity as a force so the movement of the ball shows that the rest frame of the lab isn't an inertial frame. However, in many gedankenexperiments we apply special relativity to situations on the (hypothetical) Earth with trains, etc. So there, the rest frame of the lab is silently taken to be an inertial frame.
This issue gets even more pronounced if we adopt the terminolgy of Taylor and Wheeler's Spacetime Physics which prefers the usuage of the term "free-falling frame" over "inertial frame" in special relativity. So they silently consider the rest frame of a lab on the Earth as "free-falling" even though it clearly is stationary relative to the Earth.
So my question is: Should the rest frame of a lab on the Earth be considered to be an inertial frame (= free-falling frame) in special relativity or not? Maybe there's something wrong with my association of the lab and "its own" inertial frame in the first place? (Taylor and Wheeler certainly suggest such an association by picturing a frame of reference as a grid of rods and clocks.)