What to do with students requesting deadline extension due to the death of a relative (but without a doctor's note)?
Should we just let these things go?

Yes. Either the situation is real, in which you would be causing great harm in refusing and distressing the student further OR the student is lying and they get an extra 48 hours. Do you really care if the student is getting an extra 48 hours? Is the ability to complete work within a very specific time frame one of the key learning goals of the course?