>so why wouldn't he remove the garrote and take it
>with him, like the duct tape and stun gun?

Simple. John Ramsey would not be particularly upset if he fails to see a roll of duct tape or if he does see a roll of duct tape, but that cord embedded so deeply into his daughter's tiny neck, now there is a sight! So why would he ruin that by taking the garrote with him?
That was a major part of the fun.