The graph above is a great summary of the tread depth vs adhesion issue. One of the hidden reasons why quality tyres are more expensive is that compared to cheap tyres, they have more layers of tread, each with a different compound mix. So that as the tread wears, the handling stays the same insofar as that is possible. One effect is that the inner layers can wear quicker than the outer (giving you better adhesion to compensate for shallower tread in the wet, at the cost of greater wear). So this is one reason why it's worth checking your tread at least monthly, as you may have just entered a new band of rubber that wears faster.
Deciding on this curve for a particular tyre model is a sort of zero-sum-game. And of course, as the sipes and grooves get shallower, there's no way 'round the problem that the tyre just can't shift as much water in the wet. A tyre chemist/designer will have to make a decision about how steep to make the curve, and where to flex it. So when a manufacturer says 'replace at 3mm' what they've probably done is to make the adhesion really good down to the 3mm point, but at the cost of very rapid wear and reduction in adhesion below. So this is why you get different recommendations from different manufacturers. Also, given how much poorer 3mm deep grooves are than 8mm ones in shifting water, then you might as well make that the 'flex' point where the tyre is degrading all 'round.
So this is also why claims about long-life can be disingenuous. Because you can easily make a tyre last longer by 'trading-in' some other characteristics (such as resistance to aquaplaning, tread squirm, cold temperature flex etc. etc.). The design and manufacturing skill is in getting all these things at just the right balance without the tyre costing £300 a time.
Personally, for our runabout, I take the tyres down to about 3mm and then consider replacements. If it's warm, I'll take my time looking-'round and may wait until the winters go on and then throw them away if they're below the 3mm point. For the M235i I will start looking-around to replace by the time they get to 3mm.
With winter tyres, if you're putting them on for snow, especially as part of driving in the Alps, then the change point is usually recommended to be around 4mm. Below that their snow adhesion degrades fast. If you come out of the winter at 4mm then people sometimes opt to defer the switch to summers, so as to wear the summers a bit less.
