We study the conditions of amplitude death in a network of delay-coupled limit cycle oscillators by including time-varying delay in the coupling and self-feedback. By generalizing the master stability function formalism to include variable-delay connections with high-frequency delay modulations (i.e., the distributed-delay limit), we analyze the regimes of amplitude death in a ring network of Stuart-Landau oscillators and demonstrate the superiority of the proposed method with respect to the constant delay case. The possibility of stabilizing the steady state is restricted by the odd-number property of the local node dynamics independently of the network topology and the coupling parameters.