they are still viable and still do work on the Ukraine border
They're absolutely idiotic to do so. The law is very clear on the issue - you have 90 days out of every 180 in Schengen. They may have been readmitted if they were still legal (on Day 89 for instance) - but it doesn't add an extra 90 days to the clock. There are countless people thrown out of Schengen every year for this - and bearing in mind that the standard punishment is a year long ban from Schengen, plus the permanent record on SISone4all regarding you having been an overstayer - it really is absolutely idiotic and mindless thing to do so. Perhaps the thing that confuses them is the stamp in the passport - of course, in the past, this meant another 90 days. But now, it's just a record of your entry into Schengen - it doesn't apply any time limits by itself.
I'm not surprised Americans have been doing it - they have (in general) an absolutely terrible grasp of Schengen law. The vast majority of them aren't aware (for instance) that border controls can take place absolutely anywhere in the country, and that very often, the police will consult SISone4all routinely.
The crucial thing here is that irrespective of what happens at the border, it is strictly 90 days in every 180. It doesn't matter who picks this up - if you overstay and you're found out, then you can expect trouble. It doesn't matter who finds out - it could be a French PAF guard who discovers the overstay 2 years in the future - either way, a year ban and a fine is normal.