I would say this is not the best way of thinking about it. Monthly Envelopes do experience some sort of changeover on the 1st of a new month (or the first day of a new monthly budget period, if you’re not budgeting to the 1st), but this is not the same thing as them having a due date of the 1st.
This is because a Monthly Envelope with a Due Date of the 1st and a Monthly Envelope with no set Due Date behave completely differently. In addition to the presence/lack of the black ahead/behind line, Envelopes with Due Dates track saving progress only, ignoring the current balance of the Envelope to only look at the money added in a given period, whereas Envelopes without Due Dates simply track the current balance. You can read more about this distinction in this Help Center article here.
Long story short, the addition of a Due Date does more than just stick a date on the Envelope to look at; it completely changes how the Envelope acts.
The answer to this is sort of related to my last point. Envelopes with Due Dates are intended for entirely different purposes from Envelopes without them, so they have or don’t have the black ahead/behind line to fit their goals.
For example, Monthly Envelopes without Due Dates were designed for budget items like Groceries, where you’ll likely make several purchases throughout the month. Thus the black ahead/behind line helps to track your spending throughout the month, and tells you whether you might be spending too quickly or have some room to spend, based on your budgeted amount.
On the other hand, Monthly Envelopes with Due Dates were designed for budgeted items like Rent, where you’ll most likely make a single payment in the month, and make it in full. Thus, the black ahead/behind line wouldn’t really be helpful, because it would either tell you that you’re way ahead or really overspending depending on when in the month it is and whether or not you’d made the payment yet, and that’s not really accurate. For budget items like those, you’ve either spent them or you haven’t; there’s no gradual progress to be tracked.
That example is an oversimplification, and I don’t want this to be taken as me saying there’s only one way to do things in Goodbudget, but hopefully those two answers explain things a bit better.