Your skill will accomplish
what the force of many cannot
Try catching a leaf falling... toss a bucket over the side and haul it back up...
Leaves are subject to winds which are chaotic. Sea borne litter is subject to currents and winds and is even more chaotic.
Space debris moves according to well-known and relatively simple mathematics. They can even account for atmospheric drag. Their paths are very predictible, barring collisions which do occur, but relatively infrequently due as you pointed out, the vastness of the volumes of space involved. And they are predicted, for every space mission. Launches and returns.
I also think you underestimate the mass the reflector would need to be of any use.... If you have sufficient mass to make it useful, you won't have the energy available to move it around. Make it light enough to move around, and you won't be able to do anything,
A 1.25" x 15ft x 30ft (3cm x 4.5m x 9m) titanium ellipse: Mass 4 tonnes. You could double the thickness for 8 tonnes and the shuttle could still easily get it up there. And the arm could deploy it. The Columbus science module taken up last month was 15ft x 30ft and 12.75 tonnes. It was manipulated from the cargo bay into position for attachment using a robotic arm. Albeit ISS' arm, not the shuttle's.
The reason debris is such a risk to ISS modules is because their walls are made of two skins of thin aluminium (3mm or 4mm?). Some of the debris are bits of titanium which is nearly twice as strong and 60% denser. Soft, thin collides with harder denser, and the latter wins.
If you fire a bullet at brick wall face on, it penetrates. Fire it at an angle and it bounces off. I seem to recall an incident of a gunman firing at the driver of a car through the side window at an angle, when the glass crazed but the bullet bounced off rather than penetrating. But it might have been a movie rather than a real life incident?
I'm quite expecting for the idea to be shown to be bunkum, but most of your points don't do that. Your point about the vastness of the territory is valid, but remember the idea is send the shuttle up at the end of it's useful life never to return. Remote controlled, with vastly increased manouvering fuel and no rush to get any particular place fast.
So you aim it at the point you want it to be to intercept one or more pieces of debris in a few days or weeks time and give it a gentle nudge in that direction. In a vacuum, it'll get there eventually. And you concentrate your efforts on clearing specific important orbits, like the frequently used equatorial launch paths used by things like the European ATV and Russian Progress supply vehicles to and from the ISS.
Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
"Science is about questioning the status quo. Questioning authority".
In the absence of evidence, opinion is indistinguishable from prejudice.