Saving the marriage, in the vast majority of cases, will do more good than the potential damage to all parties by dumping the marriage. How far to push to save the marriage? As far as possible, as mightily as you can.
Of course every situation is different and this is general advice. No one can direct your personal situation through an anonymous medium, especially where all the facts and nuances are unknown to anyone but yourself, other than give general pointers.
Edited – While this may be good information to tell someone who is “on the fence”, for someone who no longer has that opportunity it is only rubbing salt in an open wound.
As far as the gemorah, suffice to say the degree of mental incompetancy that precludes a husband from issuing a Get is very high and something very very few men qualify for reaching. Standard clinical depression does not come close to that level. But if it is clinical depression that is pushing him to desire a divorce, that may be a very good indication that he isn’t making this decision of sane mind. (Though it is sane enough for a Get to be halachicly sufficient.) And he should work on the depression first,after which it is IY”H cured he may realize how foolish it was for him to even want to have divorced!