1. Using the perihelion in an orbit "around the sun" as a gravity assist?: spacecraft usually care about their speed relative to the sun (characteristic energy, C3), and a (free) gravity assist around the sun won't do much. Dropping close to the sun to perform a powered bi-elliptic transfer could be a thing if you wanted to travel extreme distances (e.g. put a telescope at 500 AU to use the solar gravitational lens)

2. Using other bodies that are "around the sun" to get a gravity assist?: spacecraft do this all the time.

Also "get around the solar system faster":

1. Decrease the orbital period (lower orbits orbit faster): This is exactly what Messenger and Parker Solar Probe is doing flying by Venus/Mercury. They're 'bouncing' off of the planets, trading orbital energy and raising the planets' orbit around the sun while dropping their own.

2. Get to places faster: This is what outer planets probes (Voyagers 1/2, Cassini, New Horizons) do. If Jupiter wasn't there, these missions might not be possible.

But no one is talking about the Earth's frame of reference.

> But “the gravity of Earth” imparts no net delta-v and wouldn’t on its own allow the craft to reach Venus.

That's statement is untrue. Gravity assists with planets can provide net delta-v allowing spacecrafts to reach other planets. See the Voyager 2 gravity assists for one example.

However, the barycenter is moving relative to Venus. Imagine just the three things--the Earth, Venus, and this little object. Now imagine the object is coming almost directly from Venus, loops in a tight ellipse around the Earth, and goes shooting back almost directly towards Venus. The velocity relative to Venus changes enormously. Even if you're just concerned with the magnitude, some of the Earth - Venus relative motion gets added to the probe. Think bouncing a rubber ball against a wall that's moving towards you. The wall slows down a tiny amount, and almost all of the wall's velocity is added to the ball when it shoots back towards you.

Only in the Earth’s center-of-mass frame. In the solar system rest frame, the probe leaves with a different speed than it entered with.

> However, the probe's direction could change based on how close it passes Earth.

Both the magnitude and the direction of the velocity vector change.

This is true only in the center-of-mass frame, and that's true for all conservative forces.

If I have a bowling ball covered in springs and I throw it at a marble at speed v, the marble will traveling at a speed 2v after the collision. It would be completely correct and not misleading to say "the springs on the bowling ball accelerated the marble to 2v" even though the marble's speed in the center-of-mass frame is the same before and after the collision.

It's also true that the energy gained by the marble comes from the bowling ball in the rest frame. That doesn't make the first statement wrong or misleading. It just means that you like thinking about things in the center-of-mass frame.