Are they sure they didn't just spill some oil on the exhaust or something?

Sounds like pretty flimsy evidence on which to spend money replacing an
exhaust manifold. If the crack is big enough to emit smoke, it will be big
enough to emit noise.

That sounds like piston slap, not a manifold leak.

If I were you, I'd get a second opinion on that "cracked manifold" thing.

Smoke isn't a common indication of exhaust leaks. Although start-up smoke
can be in the exhaust from leaky valve seals, getting enough of that exhaust
out into the air would require a leak large enough to produce a terrible
roar. Small leaks normally suck air in rather than blow exhaust out.

Typical exhaust leaks from the manifold area have a distinctive sound. The
sound is a distinct tapping sound that we'd never think was an exhaust leak
unless we knew it already; it sounds exactly like a small metal part tapping
against a larger metal part. The sound is like a noisy lifter but a bit
sharper, and unlike lifters, wrist pins and rod bearings it gets noticeably
louder when accelerating. Piston slap sounds similar (including the loudness
when accelerating and quieting down as the engine warms up) but isn't as
sharp. Note that the quieting with warm-up doesn't usually apply to cracks,
just to leaks where the manifold is bolted to the head.

The only certain diagnosis is to locate the leak.

Mike