Is it possible to use random restrictions to obtain a lower-bound for $\mathsf{TC^0}$?
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 $\mathsf{TC}^0$.