These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
YMMV: Library War
Fridge Logic: The world simply does not hang together. In the era of the internet, book censoring is not going to work; the first thing the task force should do with the books is scan them and mirror internationally.
The Media Betterment Committee probably filters internet traffic to prevent anyone from accessing such content.
Another bothersome inconsistency is that both the Media Betterment Committee and the Library Defense Force are completely legal government agencies who are apparently allowed to shoot each other up on a fairly regular basis.
This is repeatedly expounded upon in the light novels, and I thought at least touched upon in the anime – one is a national government agency, the other is tied to local government.
In essence, the whole war between the Libraries and the Media Betterment Committee is a proxy war between the local prefectural governments and the national government of Japan. An equivalent might be an American federal force against a state militia. This also explains why the JSDF is sitting this out, as it's focus is to defend Japan from external attack, not intervene in a relatively bloodless civil war.
Another one is the modus operandi of MBC. Mostly we see them raiding the bookstores which is second most inefficient way of enforcing censorship (most inefficient would be raiding homes). Usually you would seek to control the publishers and/or distributors who make business by supllying bookstores (preferably by nationalizing them) and raids would come if there are rumors of samizdat being distributed or if there was a mistake in publishing process.
There are some internet posters who questioned why the LTF doesn't fight harder against the MBC, or allow them small victories. This becomes a lot more understandable when one realises that the LTF is significantly outgunned and outnumbered by the MBC, and that if they attempt to escalate, the MBC can easily ramp up their forces and curbstomp them. We see this in the final story arc of the anime, where the LTF is losing hard against the MBC in a straight up battle.
So the fight is cast as one between local and national governments. If the Library were nationalised, then the military conflict would have to end. Wasn't the cause of  a conflict between federal and states' rights?
Genius Bonus: What do you get if you convert 505 Kelvin (written "505K") to the Fahrenheit scale? 451 degrees Farenheit, which is the real title of the book they're referring to as 505K.
451 degrees F is actually closer to 506 degrees K (about 505.93 degrees K), but close enough.