That’s great for little bits of debris from time to time.
Not going to hold up long in a Kessler syndrome scenario.
And if they are travelling in opposite directions, double that.
If we can do a bit of asteroid mining, then we could probably shield spacecraft. Or, we could put a small asteroid in low Earth orbit and just use that bulk for shielding.
But, as long as we have to launch from the Earth, it’s not going to be practical to put up any sort of armor that would allow you to stay in low earth orbit for too long.
As I understand it, it has nothing to do with self-sacrifice; the shuttle could use a robot arm or something to just give the enemy satellite a push downwards towards Earth. No harm to the X-37 shuttle itself. Then fly up to other enemy satellites, rinse and repeat. In this way, you could take out entire fleets of enemy satellites.
Of course, it would only be a matter of time until China or Russia had similar shuttles that could do the same to American satellites (except for sats like GPS, those are much too far away from Earth.)
To even contemplate doing that, you’d have to first match orbit with the satellite. And the robot arm on the Shuttle isn’t suitable for “throwing” anything: It’s really, really slow. And if you can match orbits, you could just put the entire enemy satellite in your cargo bay and abscond with it: I’m sure the US intelligence services could find some way to make use of that.
EDIT: And “then fly up to other enemy satellites, rinse, and repeat” is just flat-out ludicrous. You’ve matched orbits with one satellite. Having done so, you don’t have enough fuel to match up with any more.
People have this tendency to think of “space” as one place, and so if you’re there anyway, you might as well go to all the other things in space. But space isn’t one place; it’s a whole bunch of places, almost all of them in fact. It’d be like saying “You can just get a truck, and use it to attack a target in Nebraska, and then drive to Australia and attack a target there, since you’re on Earth anyway”. Except it’s even worse, because you have to match both position and velocity.