The solid rocket boosters are toxic, but are safer and more reliable to keep fueled and ready to launch continuously than less-toxic liquid-fueled rockets, which are an available technology.
Immediate/continuous launch readiness is something of an essential feature in this role, so you don't compromise on it.
How does the toxicity of post-combustion byproducts compare between the two? Does GP mean that the exhaust of solid rocket fuel is more toxic than the exhaust of liquid rocket fuel?
That's definitely true too, given that the exhaust of most kinds of liquid rocket fuel isn't toxic at all. Hydrolox releases water, and methalox and kerolox releases water and carbon dioxide. (And that's all of the commonly used liquid fuels.)
Note that rocket engines do not completely combust their fuels, far from it. So a lot of that highly toxic solid fuel in its uncombusted form is still gonna be in the exhaust.
They do not fully combust their fuels because they aren't stoichiometric, typically. But all the fuel components are heated to high temperature, so any energetic (and often not coincidentally toxic) chemicals are reduced to a mixture in close to thermodynamic equilibrium. Hydrazine, for example, gets converted to nitrogen and hydrogen, with some of the hydrogen being oxidized.
The solid rocket exhaust will be more toxic, as it will contain hydrogen chloride (for typical composite propellants). Exhaust of the liquid rocket will be mostly innocuous (some carbon monoxide). The danger is propellants released in accidents, or when spent stages with residual propellants crash.
This particular missile's boost stage is based on the upper stages of the Minotaur-C rockets, which uses Orion rocket motors. Orion rocket motors are solid rocket boosters.
In general, solid rocket motors consist of particles of fuel and oxidizers suspended in some sort of plasticy/rubbery binder. These binders are usually some fun complex polymer. The Orion rocket motor apparently uses a form of polyurethane - which produces hydrogen cyanide gas on combustion.
Desirable properties of binders include: having a liquid form that can cure into a solid form in a controlled way (this makes manufacturing the suspension easier/possible), curing into something that is somewhat elastic/rubbery (I suspect easier handling, and more stress tolerant), and ideally also somewhat usable as a fuel (so that it's not a dead weight). I suspect that most polymers with those properties all have nasty combustion products one way or the other.
Some things to consider - most rockets, especially interceptor missiles spend most of their time NOT being fired. They are being stored and handled and tested, but not fired. That means that when considering your choices, you can't just blindly select for exhaust toxicity + rocket performance - you need to consider the operating cost and safety as well.
Its for this reason why solid motors are a go to - they are much easier and safer to handle than liquid fueled rockets throughout operation. They also have the desired properties in that they are effectively always ready to launch - you don't need to worry about managing fuel tanks or whatever.
"Command and Control" had an excellent discussion of the wide variety of ways that icbm fuel and oxidizer will kill you. (It's a truly great piece of journalistic writing, with excellent systems thinking. highly recommend if you haven't read it.)
There is a highly entertaining book by John D. Clark, titled “Ignition! An Informal History of Liquid Rocket Propellants”, which will tell you more than you ever wanted to know. You can easily find a copy online.
Basically hydrazine (nuerotoxin, carcinogen) is nasty stuff, but perfect for rockets and missiles. The military has little incentive to do anything about it.
After the second or third, I wouldn't take the alerts seriously anymore. At least they are running their psychological and behavioral analysis operations there, instead of someplace like Hawaii:
I have no idea why you're being downvoted because that's perfectly fine argument. Alert fatigue is a thing. And if a missile really was headed towards there, well, all hell would break loose.
Alert fatigue is a thing true. But these are people living within a mile a launch site that's designed and sited to protect not just Alaska, but a good chunk of the lower 48. Missiles wouldn't necessarily be targeted at the fort.
The chain of logic is straight forward if you accept the premise that it makes sense for the site to be in an alert state for all North Korean missile launches.