There are detached houses in Whitchurch for that money. Mostly to the right of the Manorway between Whitchurch and Birchgrove. But still Whitchurch. Cheaper as it’s split off the village via the dual carriage way still expensive on the dual carriage way mind! Seems main roads don’t effect price as much in Cardiff and the manor way is “The Main Road” in north Cardiff.
You’ll also get detached 1960s houses in Rhiwbina - away from the garden village: conservation areas.
And in Llanishen/Thornhill - again away from the village and they’re 1950s - 1990s houses some very cheap on the council estate and ex council estate and some expensive the nearer the village or Lisvane you are.
Heath - a rare few detached ones for your budget they’d probably need a lot of work.
For your budget you will have to compromise for a detached house. It will either have to be away from the villages or have less kerb appeal/ smaller in general.
Lots of the more central “nice” areas have very very few detached houses and the ones which are, are usually converted coach houses at the bottom of gardens.
Radyr, Cyncoed, Lisvane are mostly detached but will be over your budget (and mine)
Pontcanna, Canton, Llandaff North, Roath are mostly terraced, semi and still pretty expensive. The only detached ones. Roath Park, Llandaff city are big BIG money.
The new estates they’re building off Lisvane and Radyr have detached starting at 350k but I imagine they’d be smallish.
I wouldn’t rate any other areas of Cardiff. Not that they’re bad but they’re not nice IMHO.