Yes, but not in any sensible manner.
Space mirrors could certainly work to manage Earth temperature, the issue is that to do meaningful amounts they need to re-direct a reasonable % of the light reaching Earth, making the needed area a sizable % of earth's surface. This produces designs with areas measuring 1000s of km on a side and weighing millions of tonnes.
These designs normally assume placement at Earth/Sun L1 point. This last bit is important, since at L1 our massive and expensive structure is shading 100% of the time. Putting the same amount of 'shade' in LEO means more than half of it is either behind earth, or off to the side, needing more 'stuff' launched. There is no way to have an solar shade orbit that stays on the 'day' side*.
Designs to date also assume an active structure that keeps facing the right direction to provide shade. If uncontrolled, flat shapes will often not be shading effectively and non flat shapes will be mass inefficient - both options increasing needed mass to orbit. Worse, some shapes may stabilize end on to Earth, making optimally inefficient shades. It would be easy for the massive number of rocket launches to get millions of tonnes of shade material aloft to have more climate effect than the shade itself.
Collisions between shades is also a problem, since it will tend to convert them from something efficient into something less so. And for very large numbers of small shades collisions are inevitable due to how all orbits not in the same plane will intersect twice, and the way these objects will be acting as solar sails so will not stay in original orbit.
Which yes will make a kessler cascade of the question, but leave earth with a debris ring that may in fact make things worse! A poorly operating LEO shade system can end up hindering cooling on the night side of earth, reflect light onto earth rather than away to space or block visible light but re-radiate infra red to Earth anyway.
Further, a LEO system will tend to re-enter as on average collisions will remove orbital energy, and several millions tonnes of shade material in the upper atmosphere will be complicated to deal with.
So, while a massive deliberate Kessler event could impact the climate it certainly would not be 'control'.