The moment after a goalkeeper claims the ball is one of the best counter-attacking opportunities in a match. Whether it is used depends almost entirely on how the goalkeeper distributes.
Why the opponent is exposed at that moment
A team that has just attacked has committed players forward, and its defensive line is usually high with fullbacks advanced. Those positions do not reset instantly.
The window lasts a few seconds, during which the attacking team is at its most spread out. After that, players recover and the shape re-forms.
A goalkeeper who takes time to place the ball and kick it long has already spent the window. The ball then arrives in a contested area against a reorganized team.
The throw is faster and more accurate
Throwing releases the ball almost immediately after the catch, and the ball travels flat, arriving at a teammate's feet rather than as a contest in the air. Possession is retained rather than gambled.
Accuracy matters because the receiver is usually a wide player in a specific pocket, and a ball thrown to the wrong side of that player kills the break. A throw can be placed; a long kick cannot.
The limitation is distance, since even a strong throw covers a fraction of a goal kick. The break must be started closer to the goalkeeper.
Where the ball should go
The usual target is the fullback or winger on the side away from where the attack ended, because that side has the fewest opponents. Play is switched before pressure can arrive.
Alternatively the ball goes to a forward dropping into the vacated central space, which is available precisely because the opponent's midfield pushed forward. That option is faster but more contested.
Teams that counter well from goalkeeper possession rehearse both, with players holding designated positions during opposition attacks. The break is prepared before the save happens.
The rolled ball to a center back is a different decision
Rolling the ball to a nearby defender restarts a build-up rather than a counter, and it is correct when the opponent's shape has already recovered. It is a decision, not a default.
The error is treating it as the default in every situation, which converts every catch into slow possession and throws away the transition entirely. That habit is common at every level.
Goalkeepers who scan the field while catching are making the read before they have the ball. The scan is the coachable part.
Why the six-second rule is not the constraint
Goalkeepers are limited in how long they may control the ball with their hands, and referees have applied this more strictly in recent seasons. The practical effect is to discourage the slow reset.
For a counter-attacking team the limitation is irrelevant, since the whole point is releasing the ball immediately. The rule pushes in the same direction as the tactic.
What actually limits the throw is whether teammates have moved into receiving positions. If nobody is available, the goalkeeper has no option but the slow restart.