But I will briefly point out the flaws with your thought experiment:

1. Point-like particles don't exist, so the force will never become infinite. In fact, this is the wrong paradigm all together -- if you're talking about two particles "colliding", then what you're really referring to is their interaction cross section (which you can think of as being their probability of interacting). Neutrino (weak interaction) scattering is incredibly rare. Gravity is much weaker still. The particles would indeed pass right through each other then oscillate back and forth (not forever, but practically so).

2. You have assumed that both particles begin with a relative velocity of 0. This is not a good assumption. In the real world, two dark matter particles would have random relative velocities according to some distribution, which means they would have angular momentum relative to each other. Again, that angular momentum can only dissipate through friction. If the particles interact solely gravitationally, then they can only lose energy via gravitational waves (i.e., extremely slowly).

You probably know that the Earth formed due to gravity. But gravity is only half the answer -- you also need something to slow the matter. On Earth, that was the electromagnetic force. Purely gravitational matter has only gravity.

One might ask why shining a flashlight during broad daylight (or say while doing work around the international space station) doesn't produce black holes. Or why there aren't black holes in the first image at <https://en.wikipedia.org/wiki/Caustic_(optics)>.

Particle DM is still quantum particles, with Compton and de Broglie wavelengths. A cold milliectronvolt axion has a wavelength on the order of a kilometre. How do you trap a pair of them in a volume comparable to the Schwarzschild radius of about 10^-50 metres?

Additionally, an appreciable mass-energy of dark matter (one you can expect to be described tractably by the Raychaudhuri equations) will be a many-particle system occupying a noncompact volume.

Newton's constant G in SI units is the force in Newtons between two one-kilogram masses at rest with respect to each other at one metre apart. It's small (< 7e-11). You will wait a long time -- about a full Earth day -- for two kg of anything initial 1m apart to get very much closer to each other because of (only) their mutual gravitational interaction.

However, each of the pair of kg-mass blobs of DM particles is not likely to stick around intact that long because (by virtue of them being weakly interacting particles) nothing around them is strong enough to keep them from running away due to thermal motion. They are only cold in the sense they move slowly compared to hot dark matter in the form of relativistic neutrinos; they can still move about at many m/s, which is a lot faster than m/day. Newton's constant is rather small, and rather too tiny when thinking of coaxing a couple kg of CDM into a volume comparable to the Schwarzschild radius of about 10^-27 metres.

One might compare this with two kg-masses of water ice. In a ball of ice various interactions trap molecules within the ball even against sublimation. The ice balls will come into contact in about a day. Thus we can have things like Saturn's rings or periodic comets, helped by the non-gravitational interactions of water ice keeping the mass-energy-momentum from flying away before gravity can do its work.

Indeed, keeping galaxy and galaxy-cluster dark matter halos from flying away puts an upper bound on their temperature (if you heat up the water ice above to a couple hundred kelvins it can sublimate away in minutes). Halo DM needs to be cold in order for it to stick around.

tl;dr, two sets of non-self-interacting (collisionless) dark matter just slide right through each other. cf. the famous Bullet cluster collision. You'd need at least a galaxy worth of dark matter and at least many millions of years of Chandrasekhar dynamical friction (radiating away very low amplitude gravitational waves as structures ("subhalos") form and move through each other) to make a black hole from DM alone.

I was going to say something about how we don't really know how gravity works at the infinitesimal distance level as we do not have a quantum gravity theory, but IANAP.

With the caveat to follow, as one takes the wavelength of a photon towards zero (i.e., ultra high energy gammas), its contribution to the stress-energy tensor's expectation value at a point climbs. In turn the average radius of curvature around that point becomes smaller than the gamma's wavelength.

When we are in that regime measuring distances by wavenumber and measuring times by photon frequency gives useless results. Likewise, infrared photons with wavelengths comparable to the diameter of a black hole and finding themselves lurking around black holes, that is hard to understand too.

Now we introduce the foreshadowed clarification.

We can always find some ultraboosted observer who would (in its rest frame) measure that a photon emitted from your computer screen as an ultra high energy gamma ray. This doesn't mean your computer screen is creating small black holes. After all we can always find some other observer who sees a photon from your screen as having a wavelength of thousands of light-seconds.

So let's consider instead a photon scattering inelastically off a charged particle. The incoming photon momentum need not equal the outgoing photon momentum. Some of the momentum can be transferred to the charged particle ("recoil"), but some of the momentum can be absorbed into a composite charged particle's internal degrees of freedom. That might lead to, for example, <https://en.wikipedia.org/wiki/Photodisintegration>.

But as we take the incoming photon energy ever higher, and deposit more of it into the charged particle, we start growing the stress-energy tensor's expectation value in the region of the scattering as above. The radius of curvature shrinks, and in extremis may keep daughter products (low energy outgoing radiation ranging from the outgoing scattered photon to the results of photonuclear and photosubnuclear processes; we can also get pair production from a high-energy charged particle inverse-compton-scattering a photon to sufficient energy) from leaving the immediate region.

What happens then? Semiclassical gravity, which is quantum field theory for matter on a classical curved spacetime background, gives us confusing results because the excitations in the quantum fields don't localize like a classical theory, and our averaging procedure (to obtain the expectation value) gives us a smeared stress-energy that less and less resembles the quantum mechanical system (and consequently the generated curvature looks unphysical).

The radius of curvature and the Compton wavelengths of the particles in the strongly curved region are small, but not infinitesmial.

"Possible Connection Between Gravitation and Fundamental Length ", Mead 1964 <https://doi.org/10.1103/PhysRev.135.B849>, which you can access by pasting a Swedish scy hob prefix in front, takes the priority in the development of this line of argument. He finds that the characteristic length scale is on the order of 10^-35 metres. (cf. <https://en.wikipedia.org/wiki/Planck_units>).

But we can have two 10^-35 m gammas pass right through each other without engaging this argument. The gammas do indeed generate stress-energy, but as noted before one can always find observers who will see these gammas as infrared, and their sky won't be full of black holes in one direction and devoid of any photonuclear interactions in the other. (We can even have extreme accelerators ("Unruh observers") who count multiple gammas where non-accelerated observers see just one -- in principle acceleration can make the observer's particle count climb arbitrarily high, but that doesn't result in the observed system becoming heavy, much less a black hole).

The wavelength (and particle number) in some appropriate inertial frame of reference for the scattering detailed above is what we want when analysing this type of interaction. This is usually the "COM" frame <https://en.wikipedia.org/wiki/Center-of-momentum_frame#Speci...>, which can be calculated by arbitrary observers of our charged particle + photon (even ones who want to count more particles than an inertial observer, or redshift everything to very low energies).

tl;dr: What we're doing above is building a COM frame with a small spatial volume and high energy and asking what is the calculated curvature in that part of spacetime. That gets messy long before we have to bring in infinitesimals or start to worry about making a small black hole.