It was my wife's school friend Marjory Bacon that headed that project. (She was also the person who turned on the Hubble telescope and initiated the first self-diagnostic)
The first answer is ICBMs are, now a days, mostly an IR optical thing.
The flightpath of most Russia to US ICBMs would never have passed over the observatory, obviously, but it was a major radar technology research site in the early years.
Source? This seems unlikely since ground-based visual tracking would be obscured by e.g. clouds. Is ICBM tracking all space-based and visual? (I can see why IR is useful to detect the initial boost, but it would also seem rather useless to track a small, passive object in freefall).
https://en.wikipedia.org/wiki/Defense_Support_Program
https://www.lockheedmartin.com/content/dam/lockheed-martin/s...
That's probably anomalously hot as it plows back through the atmosphere.
I think radar stealth coatings are sufficiently advanced that they swung the balance back in favor of IR for tracking purposes.
Historically: it was hard / unknown how to make a flying object that resisted reflecting radar
Current: it's hard / unknown how to make a flying object that's resistant to radiating heat
I would assume that if you're tracking it on re-entry that's a little too late.
Launch - early warning, max time, low certainty of trajectory
Boost - trajectory and impact refinement
Re-entry - terminal maneuvering, impact prediction, terminal interceptor cueing
I'd imagine you'd have ground-based sensors for terminal interception refinement, but having a wide angle view to cover highly maneuverable glide vehicles would be useful too.
There are techniques to reduce infrared signature, though:
https://en.wikipedia.org/wiki/Northrop_Grumman_B-2_Spirit#In...
Just a second of observing the plume's trajectory will tell the difference between and orbital or ballistic one. Ballistic missiles will also be launched over the poles to travel the shortest distance to their targets. So IR plumes over the North Pole either means Santa's upped his present delivery game or there's an incoming ballistic missile attack.
Short story: before our nuke strategy shifted to basically all ICBM's and submarines, the plan was to nuke the soviets with fleets of long range bombers. To get the bombers in, though, you needed to know where the air defense radars were. The U2's were good at flying too stealthily and too high for missiles, but not great... one or two did go down creating some, er, light international incidents. So eventually someone figured out big air defense radars bounce off the moon when the moon is overhead, and suddenly there was a lot of CIA money funding radio telescopes like Arecibo and the Stanford dish.