The real answer: It's entirely beyond our capabilities and not achievable at all.

Not true. Well timed gravitational slingshots have been used on many existing missions. e.g. https://en.wikipedia.org/wiki/Parker_Solar_Probe

Of stable masses capable of analysis in advance to their mechanical properties under thrust. Asteroids are known solely by their orbital elements and we have literally no idea how the specific mass works as an aggregate or bonded entity, under load or stress.

So strap a rocket to one and find out. If it doesn't work grab another.

Having created literal mayhem in an unstable orbit of uncounted masses, ie for a giant externality. Next flight up, don't be within a large amount of space nearby in orbital terms, without care. (Maybe. Im not a kerbal person or an astronomer or flight engineer. This is a precautionary question to risk and to whom it applies)

Asteroids are very far apart from each other on the whole. Kicking one off in a different direction with a rocket isn't so different from a change in direction from a (rare) collision, except that it will almost certainly only affect one asteroid. ~25% of all the mass in the asteroid belt is the dwarf planet Ceres, which is only 0.0128 the mass of our moon. https://en.wikipedia.org/wiki/Ceres_(dwarf_planet)

There’s a vast amount of rubble flying about any which way up there already. About 100 tons of rock and dust enter the earths atmosphere every day, it’s just that space is so vast the chances of any of it hitting a vehicle are quite small.

You can line up a slingshot with a very low rate of acceleration, though.

Yes. I probably overstated my case. How quickly would that deliver a benefit?

It could take years for each mission, but of course at industrial scale you could have a continuous delivery queue and processing each asteroid could provide months or years worth of material. Think of each asteroid as a new mine being opened up. These initial missions are just proof of concept to test out techniques. You have to start somewhere.

The energy cost of changing something's orbit is not just mass times distance, otherwise launching a 1 metric tonne probe to Mars would have been impossible. Thanks to orbital dynamics, there are a lot of opportunities to get huge bang for your delta-V buck.