You bring up some great points. You're right, there are many things you cannot do without the death certificate. On top of that, there are things you cannot do until an executor is named.
The timing sometimes can't be controlled. My mom's probate lasted 18 months and required us to file a tax return twice.
And yes, I neglected to mention credit reports. We did this as soon as the trouble began since it can also help keep people from opening accounts in the deceased's name.