Containment of ACC$^0$ in TC$^0$ The Complexity Zoo states that ACC$^0$ is contained in TC$^0$ and links to the paper On ACC and Threshold Circuits.  However, what the linked paper proves is that depth-3 threshold circuits of quasipolynomial size can simulate ACC$^0$ circuits.

Wouldn't it be necessary to prove that constant depth threshold circuits of polynomial size simulate them rather than quasipolynomial?  Is it obvious how to do the simulation with just a polynomial number of gates?
You can show directly how to simulate a polynomial fan-in modular gate using polynomially many threshold gates. The idea is that given inputs $x_1,\ldots,x_m$, you can use two threshold gates and an AND gate to determine whether $x_1 + \cdots + x_m = w$ for any $w$. Taking at most $m+1$ copies of this gadget and ORing them, we can simulate a single modular gate.