Umm, no. Maximum energy gain for a gravitational assist is a slingshot maneuver where you narrowly miss a head on collision with the planet, you will then be whipped around on a parabolic path and depart in the opposite direction with twice the planet's velocity added to your own. The "gravitational tugboat" maneuver you describe is great for minor boosts and course corrections, but is unlikely to be used for speed unless a slingshot maneuver is incompatible with reaching the desired destination.
As for a solar slingshot, yeah it's pretty pointless for in-system travel - it's hard to get close (not to mention survive the passing), and since it's basically the "stationary point" for the solar system you can't steal much speed from it, so once you reach your starting distance you'll have roughly* the same velocity as when you started with. Unless you just want to briefly go really fast for some reason, or are on an interstellar vessel seeking a gravity assist on your way to somewhere else in the galaxy, the sun is pretty useless for gravity boosts.
* You won't leave a solar slingshot with exactly the same velocity because the sun itself is orbiting the solar-system's barycenter, typically between about 1/2 and 1 solar-diameter from the sun's center and constantly moving as the orbiting of the outer planets shift the system's center of mass. So there will be some velocity transfer, just not enough to be actually useful.