Re: I would have thought in modern warship design you start with the weapons....
Nope. You start with the role. In the case of frigates that's anti-submarine, with a side order of anti-ship and general patrol.
Then you pick the weapons you really want. Then you take off a few when it all looks too expensive. Then you get derive ship design.
So ASW means helicopters and a towed array sonar. Plus torpedoes, decent point defence AA and some anti-ship capability.
I think the idea of building some cheaper ships is that the RN do lots of things that don't require the specialities. So they traditionally always have a ship in the Carribbean that does anti-drugs patrols, goodwill visits, and is around to help with support after hurricanes. That doesn't need to be state of the art. There are often ships wandering round Africa, doing similar, when they're not doing anti-pirate duty off Somalia.
It's a balance of having too few assets, against having a bunch of lower quality ones that make you look strong on paper, but are no actual use if there's a real need for them. So for example if we ever had to regularly start patrolling the Persian Gulf again, you ideally should't send ships in there that don't have decent anti-missile defence, as Iran has lots of anti-ship missiles - and might let a few off. But if tensions are below actual war, then a few hulls to fly the flag and keep off Revolutionary Guards speedboats are all that's ever been required in the past. Plus minesweepers.