Is it possible to use random restrictions to obtain a lower-bound for $\mathsf{TC^0}$? There are several well-known $\mathsf{AC^0}$ circuit size lower-bound results based on random restrictions and the Switching Lemma.

Can we develop a Switching Lemma result to prove a size lower-bound for $\mathsf{TC^0}$ circuits (similar to the lower-bound proofs for $\mathsf{AC^0}$)? 

Or is there any inherent obstacle to using this approach for proving $\mathsf{TC^0}$ lower-bounds? 

Do barrier results like Natural Proofs say anything regarding using Switching Lemma like techniques to prove $\mathsf{TC^0}$ lower-bounds?
It is actually possible to make use of random restrictions to prove lower bounds for threshold circuits.

In particular in the paper Size-Depth Tradeoffs for Threshold Circuits, Impagliazzo, Paturi, and Saks use random restrictions to prove a superliner lower bound (on the number of wires) for constant depth threshold circuits computing the parity function.

With regards to proving superpolynomial lower bounds for $\mathsf{TC}^0$
 circuits then yes, the natural proof concept is of relevance since there are constructions of pseudo-random function generators in $\mathsf{TC}^0$.