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.