Space elevators are OK for planets that spin fairly fast & aren't too large. Earth is marginal for a space elevator of even the strongest possible material.

An ORBITAL RING would be far better. https://www.youtube.com/watch?v=LMbI6sk-62E&t=4s

If you're trying to break up a planet for mining, you need to overcome its gravitational binding energy. It doesn't really matter how you do it -- blowing it up, lifting it piece by piece, vaporizing with a giant mirror -- you need to pay this energy cost to get rid of the gravity well. For an Earth-sized planet, this is roughly $2*10^{32} J$, or about seven days of solar output. It's a good thing you've got a Type II civilization -- this is an expensive proposition even by your standards.

As a slower alternative, you could send the planet on a close flyby of a gas giant, within the Roche limit, to break it up through tidal forces. This is cheaper: for Earth, it would only require about $3*10^{31} J$, one day of solar output, to change the orbit, with the breakup energy coming from the gas giant's kinetic energy.

Solar concentrator.

Build a big dish at the Lagrange point and heat it up using the power of its own star. The beauty of this method is that by controlling the heat, you can first vaporize off any elements you are not interested. Then carefully turn up the heat and vaporize off the elements you are interested in and capture them. This combines collection with purification (or more correctly, fractional distillation), and saves you schlepping impure materials to some other site for purification.

Freeman Dyson (the man who conceptualized the Dyson Sphere) had thought about this, and created the Dyson Planetary Spin Motor to break planets apart (this has nothing to do with the other Dyson who makes vacuum cleaners).

Dyson Planetary Spin Motor

Essentially, you turn the planet into a rotor by wrapping the planet with a metal grid and energizing it. For a planet like Earth, the rotation speed could be doubled in about 2500 years, and the planet accelerated to the breaking point in @ 40,000 years. Obviousy, you could gain efficiencies through various means, including using a superconducting grid instead of a metal one, and using orbital power stations to beam far more energy into the grid.

This would likely work on any rocky planet, and does not need conveinient other planets to collide with - even a single planet orbiting alone can be dissasembled this way.

If you do have more planets, another writer named Paul Birch had some interesting ideas. Using a device called a "light sail windmill" in near solar orbit to capture the energy of the Sun, he proposed to use streams of high speed particles to change the orbits of planets, or even their spin rates (the main idea was to spin up Venus so it would have a 24 hr day and put in a co orbital position to Earth). The details can be found in his papers here:

https://orionsarm.com/fm_store/MoveAPlanet.pdf

https://www.orionsarm.com/fm_store/SpinAPlanet.pdf

Since he is speaking of using far more energy than Dyson (about 0.2% of the Sun's radient energy for 30 years), the time frames are much shorter - a 30 year project to move Venus would be within the working lifespan of an engineer today. Moving planets rapidly around the solar system provides many opportunities to crash them into each other, use gravitational "slingshotting" to adjust the orbits of multiple planets, or, since the stream of pellets coming from the light sail windmill are carrying so much energy, use them directly like a sort of cosmic chainsaw to carve up the planet (the pellet stream cannot be reused if you do this).

So here are two theoretical methods to mine entire planets for material.

Use Von Neumann swarms to build power plants, refineries, and magnetically accelerated launch platforms and systematically deconstruct the planet. Material payloads would be intercepted by orbital platforms and processed or rerouted accordingly.

The biggest hurdles to accomplish this are material availability and energy consumption.

Powering an O'Neil cylinder across interstellar distances probably requires fusion reactors, so we'll assume your tech level has mature fusion reactors. If not, then a combination of fission and solar panels will have to suffice to power our mining operation (and probably slow it down a few orders of magnitude).

Step 1: Locate metal rich asteroids with low gravity and deconstruct it to create a seed swarm of Von Neumann probes and processing space stations.

Step 2: Survey the planet for potential mineral deposits and disperse the probes accordingly.

Step 3: Harvest materials and construct power plants, processing facilities, and factories for more probes. Expand across the planet.

Step 4: Construct magnetic acceleration platforms (coil guns / rail guns), bundle valuable resources into launch vehicles, and hurl them into space.

Step 5: Intercept the bundles at apogee with your orbiting stations and redirect them to your orbiting ship yards.

Step 6: Watch your autonomous swarms continue to devour the planet and churn out more spaceships already in orbit.

Not sure if this helps any, but here it is-

in one of the star wars books i read once(I can't remember which one) we learn who built the death star. they had no idea it was built for destroying planets, but just thought it was for mining. the death star would be used on dying planets to allow the interior metals and resources to be easily accessed rather than digging all the way through the planet. of course, such a station would be nearly impossible, but i think it might help with your mining. just be careful not to point it at people or mess up orbits or anything.

Centralize your processing

You could put all of your mineral processing equipment in one place and bring those pesky rocky planets to the processing center. On Earth, that's the strong preference. Ships, cars, and other movable objects are brought to centralized places to be broken down.

Your probes could move the planets with a series of solar sails as described in the journal Astrophysics and Space Science. As each planet arrives, it would smash into whatever planets are already there, breaking it down into smaller pieces. Your probes could then chew on these smaller pieces and convert them into materials. Whatever you want to do with those materials could be located in the same solar system, which would simplify your probes.

I was trying to work this out and the best solution I could think of on my own is the oldest technology ever: banging rocks together. As you suggested:

What about destabilizing the orbit of a moon and smashing it into the planet to "liberate" some of its mass?

But let's increase the scope of things here.

Boffins believe that the Moon formed when a planet the size of Mars collided with Earth some billions of years. The debris from both planets went into the Earth's orbit and after some unspecified amount of time coalesced into the Moon.

So what you really need is two rocky planets of about Earth size. Throw them against each other. The planets will mostly disassemble and you will get a lot of raw material floating in orbit of the star.

To illustrate how it would go: here is a video from the University of Bristol simulating two super Earths (i.e.: two rocky planets with each having the mass of 10 Earths) colliding head on: https://www.youtube.com/watch?v=0GvfoD-UW5A. The resulting debris would have the gravity of both planets, but you could easily scoop the material in flybys.

How long would it take for our fleet of probes to accomplish this?

Depends on the technology used to move the planets. You could either use gravity slings (could take millions of years) or you could move a gas giant by force and have it drag the rocky one around (could take millenia).