This has nothing to do with coherence or the fundamental physics distinction between "normal" and laser light.
It's simply a question of finding a source that is intense enough to induce a fine cutting edge. That is, the source must be both powerful and one must be able to concentrate it into a very small spot. Cutting happens when there is highly intense local heating in a very small area of the sample.
The mechanism of stimulated emission allows the generation of huge amounts of light all in exactly the same momentum state. What this means is that the output is high power and very nearly a plane wave, with a low aberration wavefront. Such a wave can be focussed to near to a diffraction limited spot. Thus stimulated emission enables both the fundamental requirements of power and low wavefront aberration, equating in this case to high ability for concentration.
At $10{\rm \mu m}$ wavelength, that of a ${\rm CO}_2$ industrial machining laser, that implies a spot size of about $20{\rm \mu m}$ focused through a $0.3NA$ optical system. With thousands of watts continuously output, this equates to an intensity of terawatts per square meter at the "cutting edge".
In contrast, the Sun is not a collimated source - it is an extended one. The best you can do is focus it down to a tiny image of the Sun. Let's do our calculation for a 0.3NA lens one meter across. The distance to the sample is then about 3 meters. The image of the Sun is then $\frac{3}{1.5\times10^{11}} \times 5\times 10^8$ meters across, or about one centimeter across. Through our one meter lens, we get about $600{\rm W}$. So we get about the same power as in our laser example (somewhat less) through an area that is $\left(\frac{0.01}{2\times 10^{-5}}\right)^2 =2.5\times 10^5$ times as large. Our intensity is thus five or six orders of magnitude less than in the laser example.
There is limited ability to improve this situation with a bigger lens; as the lens gets wider, you need to set it back further from the target, with the result that the area of the Sun image grows at the same rate as the area of the lens, and thus the input power. The intensity stays roughly the same.
LEDs
The OP also asks about LEDs. Although modern LEDs can output amazing powers, they, like the Sun, are also an extended source, comprising a significant area of highly divergent point sources, so the light output has a high étendue and cannot be concentrated into a tight spot. This highest power LEDs needfully have a large area semiconductor chip whence the emission comes. In a laser cavity, it is also true that the first seem emissions are also highly divergent, and the first pass through the gain medium produces an amplified spherical wavefront. However, the design of the resonant cavity means that only a small, on-axis section of that spherical wave bounces back into the cavity, most of it is lost. On the second pass, we have an amplified, lower curvature wavefront; most of this is lost at the other end of the cavity too. During the first few passes, therefore, the process is quite inefficient, but on each bounce the wavefront gets flatter and flatter as only light components directed accurately along the cavity axis can stay in the cavity and the efficiency of recirculation swiftly increase. Through this mechanism of resonance, therefore, the stimulated emission process is restricted to only the most on-axis components of the light. Thus, the combined mechanisms of resonance and stimulated emission co-ordinate the whole wave so that ultimately it is a plane wave, propagating back and forth in a cavity, spread over a relatively wide cross section so that heat loading from any losses are not damaging to the cavity. This near-zero étendue, low aberration field is easily focused to a diffraction limited spot.
Solar Furnace
User Martin Beckett gives the example of the Odeillo solar furnace:
You can however use lots of lenses (or mirrors)
This is in keeping with my solar lens example above. A solar furnace is great for furnace applications, such as mass energy production or smelting. But the focused light lacks the intensity needed for cutting. The intensity in this example is about the same as for our one meter mirror. The furnace focuses several megawatts through a 40cm diameter focus, and a a few megawatts through a 40cm focus is about the same intensity as one kilowatt through a 1cm wide focus, which is what we had for our solar lens example.
