Paper invoices may be an exception, but any business doing that regularly would already have a much more cost-effective solution in place.
One possibility that comes to mind is to remain anonymous by sending through you.
Not knocking the idea, just hoping to help narrow the question of "who needs this?"
Some of our inspiration came from a couple of earlier services we built for print and mail, where we constantly fielded questions from customers who wanted their mail to arrive faster.
We'd be interested in building an integration to automate this if you're up for it. Email team@sendovernightmail.com.
However there are a ton of cases where physical mail is essentially a legal or policy requirement. Some examples are settlement notices (for class actions and recalls), construction notices (for demolitions), gym membership cancellation letters, insurance claims, credit bureau settlements, certain tax documents & filings, etc.
In most of those cases, by the time you hit the use case, you haven't been to the post office or to a Fedex location in ages. We just make it easy to get over that hump.