How come the Sun does not make the Moon crash on the Earth?
The Sun's gravity does perturb the Moon's orbit but more subtly than you imagined.
The Moon's perigee and apogee migrate eastward in an 8.9 year cycle called apsidal precession.
Also the plane of the lunar orbit, inclined about 5.1 degrees to Earth's orbit, shifts westward in an 18.6 year cycle called nodal precession.
Lunar theory also addresses various effects on shorter timescales.

If the Moon were 4 times farther from the Earth (outside the Earth's Hill sphere), then the Sun would destabilize the lunar orbit.
Even so, the Moon would most likely drift into an independent orbit around the Sun, with little risk of collision with the Earth.