It would be nice if everyone was ehrliche and comes to (and listens to) Bais Din but unfortunately that’s not the case.
From the three general categories that I believe the heter was created for:
1. The wife has simply disappeared (i.e. her whereabout are unknown),
2. The wife is in a coma or mentally unfit to accept a get and
3. The wife simply refuses to accept the get,
The first two are negated once the wife competently walks into Bais Din and the third one is negated if you’re under the assumption that she’s ehrliche, wants to do the right thing and is going to follow Bais Din.
So given that we’re discussing the heter we’re under the assumption that both parties are not sitting in front of Bais Din as Ehrliche Yidden waiting for a Psak and fully intent on following it.
If the wife is fully competent and being obnoxious it’s easier for Bais Din if she refuses to show up than if she shows up, pretends to be the victim while spewing forth an infinite amount of sheker. In the former case the Bais Din will see who the aggressor is while in the latter case the Bais Din will have to sort through all the information and hopefully figure it out. That’s why I think her complying with the summons makes things more complicated for Bais Din.
If your starting assumption is that both parties are ehrliche Yidden, once the Bais Din tells him he can not divorce her- that’s the end of it- they’ll both leave and hopefully work things out on their own. An ehrliche person will not go ahead and find another Bais Din if they don’t like the ruling. The first Bais Din will probably not write up a get for him and give him the go ahead to start getting 100 signatures.