It’s not ideal but No due date status was non-sense as there were a few places where due date (if it wasn’t entered) was assumed to be the same date as issue date.
Also having No due date status indicated it’s OK to have no due date and users were issuing invoices when they should have really issued quote or pro-forma invoice. Now that due date treatment is a bit different, it nudges certain users in the right direction (e.g. don’t issue invoice if the date of payment is “optional” and there is no due date - then it’s not an invoice).
As for my suggestion, I think it’s workable. If you don’t set due date manually, it will be some extraordinary high number which you will notice. Then you can fix it so the books are not work of fiction as you say.