I guess you had your answers already. It's a company policy. Let me put you in their shoes...
You own a company and you hired an employee. You gave that employee the opportunity to take the leave because you understand the situation he/she is up to since it's an emergency. And then that employee is not coming back. As per your policy, we say there should be 30 days notice, but that employee did not grant those. Ok, let us say he/she returned all the company property, but was not able to properly handover the previous jobs to any person or any replacement. Now, would you say that it is fair to pay the remaining dues/settlements when in fact there were no proper out-processing? In other companies, if notice is not provided by employee, that employee has to pay in lieu of that notice period. Set aside the word "fairness".. you just need to follow the policy.
Just to explain further, a notice of 30 days or more is for the HR to properly handover the works to another person until the new replacement comes. It gives HR the proper time and ways so that any jobs due won't be difficult for the company. There should be a formal out-processing --return of company property, exit interviews, proper handover of works, final dues/settlements, employment certificate and no objection certificate.
Now, what would be the consequences if you haven't done the proper out-processing? --No final settlements, no employment certificate. Worst, you may need to pay back that notice period as compensation to the company.
The only solution, you have to come back, do a proper out-processing. Put something on your resume/cv that you can be proud of. Some companies call your previous employer to know about you and how you did well during your exit.
Always remember, in any jobs that you do, treat that as your own company --take the ownership and always do a formal exit. In future, you wouldn't know when you decide to comeback. If you had a proper exit, they might ask you to return. Lastly, proper resignation is not done through emails. Even when we say that we keep the communications open, still we need to be professional. The interview is normally done in face-to-face, the resignation or any types of separation should be the same.
Best of luck Kabayan!