Tacitus says Christus suffered the extreme penalty. He never says crucifixion. Isn’t the method of death being read into a vague phrase?

On its own the phrase is vague. But the extreme penalty inflicted by a Roman governor on a provincial non-citizen accused of sedition was crucifixion in nearly every case; Roman law reserved beheading for citizens. Add every Christian source, which is unanimous on crucifixion, and the fact that a crucified messiah was an embarrassment no one would invent, and the method is not seriously in doubt. Tacitus is cited for the execution under Pilate, which is exactly what he supplies.