It takes more than a little push to drop a probe to Mercury. A 1 AU x .387 AU heliocentric orbit would be moving 22.25 km/s at aphelion. Earth moves about 30 km/s. So Vinfinity for Trans Mercury Insertion would be 7.78 km/s. From LEO the burn would be about be about 5.5 km/s.
At perihelion the ship would be traveling 57.5 km/s wrt sun. It'd be traveling a hyperbolic path wrt to mercury. Vinfinity wrt to Mercury is about 9.2 km/s. If Mercury periapsis were 0 km from Mercury's surface, turning angle of hyperbola would be about 10.2 degrees.
I think the Mercury gravity assist would be good for 1.7 km/s at best. Which wouldn't be sufficient to boost aphelion to Mars, much less Jupiter.
From LEO, Trans Jupiter Insertion is about 6.3 km/s
Regarding a gas giant's gravity capturing the probe: An earth to Jupiter Hohmann would be a hyperbola wrt to Jupiter. Vinfinity wrt to Jupiter would be about 6 km/s. From Jupiter's view point, it'd see the distant probe approaching at 6 km/s. It'd accelerate as it fell into Jupiter's well, then after Jupiter periapsis as it climbs up the well. Eventually it'd slow to 6 km/s wrt to Jupiter. From Jupiter's point of view, the probe would have the same velocity outgoing as it does incoming. It is with regard to the sun that Jupiter's gravity changes the velocity.
Jupiter capture might be possible through one of Jupiter's moons changing the probes path, or maybe if the probe's periapsis passes through Jupiter's atmosphere. But Jupiter's gravity alone would not change a hyperbola to an ellipse.
Here's a pic showing how a hyperbola's velocity approaches Vinf as it grows more distance from the central body: