There is no question that girth-hitching weakens slings considerably. But there is uncertainty about what message to extract about breaking dyneema in simulated factor-2 falls with steel weights. It is unknown whether a human body on the end of the rope rather than a bunch of steel plates would have produced anything close to the same effect, and there are reasons to suspect that maybe not. As realistic test dummies become cheaper and more available, we may get a much better idea about the practical realities of sling breakage under shock loads.
Whenever these discussions come up, it is always mentioned that you shouldn't be taking high fall factor falls on a tether anyway, and doing so constitutes pilot error rather than a failure of a protection system never intended for such eventualities. The fact remains that there are situations in which a climber might have to make a move or two above an anchor they are tethered to. The pilot-error position says that those are "leader must not fall" conditions and the climber should view themselves as being unprotected.
Certainly, if you are just going to lean back on your tether, girth-hitched dyneema slings are not going to be an issue, and this may cover most of the situations in which climbers fabricate and use a tether. If, on the other hand, circumstances will require you to climb above your anchor while tethered to it, the prudent position, even if it eventually turns out to be incorrect, would be that you have entered the solo climbing zone and should behave accordingly.
If you do a lot of tethering and are not bothered by the extra bulk of an installed tether, than I agree with Chip that the Sterling Chain reactor (and not the Metolius PAS) is the most sensible choice.
In any case, when anchoring the belayer, the rope and not any kind of tether should be the primary connection.