First, what “many objects along the way”? There are basically none. To do this requires many detours that significantly increase the distance you’re having to travel.

Second, the faster you’re going, the less useful a gravitational slingshot is because you spend less time in the object’s gravity well.

It’s a fun and useful trick within our solar system, but I don’t think the physics plays out well for interstellar travel.

Stars is what I meant. Although upon further thought I think you're right. I guess the intervening stars are unlikely to be moving at helpful velocities for such a game of pinball, metric expansion being what it is.