As I see it, the Original Poster (OP) is making a sincere and diligent attempt to comply with the Schengen regulations about overstaying.
The rule itself is fundamentally straight-forward… Briefly a person becomes an overstayer on their 91st consecutive day inside the zone. The day count is measured from the present day and extending back for 180 days.
The OP is struggling because he essentially has something ass-about-face. He asks…
How do we begin a new 180 days?
And the answer is: you don’t. What’s happened here is that the OP has it that the 180 period is a forward looking time window. So from that viewpoint it seems natural that a person can ‘renew’ their 180 days and thereby spend more time in the zone. But it doesn’t work like that. You cannot ‘renew’ or ‘refresh’ it.
As alluded to above, the 180 day period is a backward-looking time window. For any given day, you look backward into the previous 180 days (present day included) and count the number of days you were in the Schengen zone. If that number exceeds 90, you’re an overstayer plain and simple. This rule holds true even for non-visa nationals like Americans, Canadians, and so on.
Schengen overstayers get caught on their way out when they encounter an exit check. Sometimes the authorities don’t notice and other times they do. If a person gets caught, the authorities have a menu where they can select any of the following: scolding, fines, incarceration, and expulsion.
A possible source of confusion lies in the way other regulatory schemes work, like the UK. There, when your ‘visa’ expires, you need to ‘channel hop’ to get a new 6 months (as long as you can pass the landing interview).