I'm not entirely convinced they are Combat spells. "Based on" does not mean "the same as". Are there any other Duration: S Combat spells? Have you considered the type of every single other spell in Hard Targets? (that would be Manipulation)
Yes; Shadow Spells, page 16, Flame Burst (Combat; Direct, Elemental):
Type: P, Range: S (A), Damage: P,
Duration: S, Drain: F + 1
This spell sends a pulse of flame at the start of the caster's first action once every Combat Turn as long as it is sustained, in every direction, centered on the caster, and damages everything in the AOE except the caster.
However I do agree that the Spellblades are more likely to be Manipulation spells than Combat spells. Elemental Combat spells like Flamethrower and Lightning bolt used to be Manipulations back in the day of SR3 (and maybe earlier).