Schengen allows visa-free tourist travel up to a certain number of days per year. The way this is enforced is dated stamps on entry and exit, which are (at least in principle, though others have suggested UK citizens generally only get a cursory glance) checked at future border crossings to ensure the time hasn’t been exceeded. So any stamp-free mechanism to cross the border would have to create a record viewable by staff at every border in Schengen, which is a substantial IT investment.
And they are doing this! That’s the EES (possibly also ETIAS, not sure about the exact role of the two) that people are talking about elsewhere in the thread.
Waiting for the new system is likely what will happen, albeit it just seems ludicrous that stamps are still needed in 2023!
It sounds as though the issue is on the French/EU side, so I’d still be interested to understand what is different about the UK’s approach that means these problems seemingly aren’t experienced by inbound passengers.