The Netherlands (Zeven Provincien) and UK (Type 45) have recently comissioned AAW 6000t+ destroyers. So big ships are definetly not a dead concept.
The question here is whether such 6000t ships can indeed do so much more than a 3000t one? Zeven Provincien radar has a range of 2000 km... for what? 2000 km is a distance from The Netherlands to the St Petersburg (Russia) (theoretically, since there is land in the way). What can be accomplished on such a distance? Only strategic missile warfare (ZP will have SM-3s), but even that is limited if all you have are several destroyers. It is not a substitute for a missile shield, merely a partial layer.
In a naval combat a useful engagement/detection envelope is about 400km. Beyond that few missiles can engage, and thou target detection is possible, its identification is far more difficult. Both frigates and destroyers can achieve that.
A modern 6000t ship (destroyer) is as much use as a 3000t one (frigate). They carry same missile complements (Aster 15/30, SM-2, Harpoon/Exocet), similar number of torpedo launchers (2 or 4), guns are standard (76 or 127mm), and both have comparable CIWS with missiles (systems like ESSM, RAM, Aster 15) and guns CIWS (Goalkeeper, Phalanx, etc), even similar aircraft complement (standard 1 helicopter hangar). Some even have comparable endurance and range (Formidable frigate and ZP destroyer have both about 8000km), and sea worthiness. Smaller ships can even mount an impressive array of EW, like La Fayette class.
So the main difference that persists is that 6000t destroyers can carry larger marine complement or some cargo. But these functions are better carried out by dedicated ships (or sth like LCS), not a proper warship.
Additional difference is that destroyers can have more of everything. 40 vertical launchers instead of 16, and more ASW or CISW ammunition. But that has not been a limiting factor in combat for a while.
There is certainly some truth in the argument. The Formidable also carries 32 vertical cells which carry a mix of Aster 15/30. Such a frigate seems to be a sensible choice for most medium "western" navies. They should also accomodate Tactical UAV like the Scaneagle with ease, which will become irreplacable assets for future navies. Of course the larger ship will have a higher performance but who cost-effective is it?
So I wonder how much is the cost difference between, let us say a Formidable class ship and an Kongō class destroyer?