Moiraine dissipated the residues of a ward weave she had let go (the weave was no longer active, Moiraine had untied it and let it unravel on its own normally, only its residues remained and would have faded out naturally with time). There are people with the Talent to read those, and the residues can also attract the Fades.

You can't dissipate the residues of the weave for a Gateway for very obvious reasons: you have to let go of the weave, ie: in this case, let the Gateway close. You're at arrival point and there are residues left at your starting point. If someone there can read residues, they can copy your exact weave and follow (or in Elayne's specific case, a damane could have learned Travelling from the residues)

What the WO are doing is rather destroying an active weave by picking out of the weave each of the threads that form it, and letting them dissipate. Instead of dissipating the residues as Moiraine did, that leaves no readable residue of the weave at all (that may leave residues of saidar, but the weave they originally formed would be impossible to figure out).

The Aes Sedai know about that. Like Moridin/the AOL Aes Sedai they believe it's sheer madness to use that technique as it's terribly difficult, and increasingly so as you progress, to hold the weave once you've started picking out threads, that once you've started you absolutely have to finish the job, and if you lose control there's no predicting what's left of the active weave will turn into and produce... pretty much what happened to Elayne, it nearly killed her and had such destructive effects the Seanchan believed this to be a secret weapon of mass destruction. The White Tower teach the novices early to never, ever even try to unravel a weave thread by thread, that you need to let go of the whole thing, let it unravel itself. The WO rather teach how to do it with very basic weaves first and gradually work your way to more complex ones.