You are to be commended for trying to stay neutral. She is going through a lot - and so is he - and tempers flare. You could try writing her a note - so that you can get your thoughts down without her interrupting - and re-apologize for the remark that upset her. You could clarify that you want to support her in any way, remaining open to people who wish to work out their problems through counseling or mediation, and to those who feel it is better to dissolve the marriage. You told her in the beginning that she could count on you, which I'm sure was sincere. It's possible that a LOT of people said the same thing, but didn't really mean it. So she may have been disappointed by others, and put you in the same category.
Also explain that you hoped she would contact you if she wanted to talk, but you are sorry and can see that she might interpret it as disinterest, which you regret. (This is NOT to say that you really did anything wrong, but she needs to hear the apology now. You can apologize for not being clearer, without implying that what you did was terrible.) You can also say that you didn't understand that she wanted you to be more involved - you can say that her distance was interpreted by you as a desire for privacy, and you didn't want to pry or interfere.
Your own experience with divorce, of course, affects your view. You said that you feel that many people give up too easily or get divorced too quickly. However, you don't really know how long these people have had problems and how long they have been thinking about it - you only know how long they have told others about it. So maybe they are NOT doing this too quickly.
However, you DO have a sticky situation in that you want to be supportive to the husband, with whom you and your husband have been friendly. It may be impossible to do this. She wants you to know the whole story (which means saying bad stuff about him), and I don't know what HE wants you to know. There probably ARE 2 sides to it, but each of them is going to tell you their own side, which is going to make the other spouse look bad. It's up to you how involved you get. The more calm and "listened to" both parents can feel, the better it is for their child. However, if you are seen talking to one, the other may get upset. If one says to the other that you agree with their side, it will hurt you. Tread carefully.
There are lots of divorce mediators who make the process easier and more civil and fair, without trying to change the outcome. They are cheaper than lawyers. Even if they ARE lawyers themselves, they meet with both spouses so it's only one lawyer instead of two. You may want to suggest this in support of the financial picture, and it may also be less stressful. But I'd start out listening to your friend as much as you can, so she feels supported and becomes less angry. If it's too much though, you may have to impose some distance.
Good luck.