Rust wouldn't necessarily make it more responsive. It is more oriented towards safety and robustness.

Cosmic might be more responsive / efficient due to the fact that it's a new development and they can choose to implement things better and not carry old baggage, but that's about it. Rust doesn't have much say in it.

Edit: Although, if they are moving away from JavaScript in Gnome as their shell language to pure Rust in Cosmic, then you would probably see some responsiveness / efficiency gains, yes.