Why battlecruiser speed justified the investment

Were battlecruisers generally worth their cost despite some bad designs?

Yes for their era, because speed could control an engagement until aircraft and missiles displaced that advantage.

Battlecruisers were rational products of an era when a ship's speed could determine whether it accepted, escaped, or shaped an engagement. That operational freedom could justify the investment even though individual designs had serious faults. Aircraft and missiles changed the calculation because their speed and reach, rather than a ship's extra knots, increasingly set engagement range. A hypothetical 300-knot ship shows how that calculation could change again: the question would be whether a missile could reach the ship before exhausting its range.