Apologize (sincerely)
Edit: Response to comments
First, an overall clarification of the focus of my answer. I'll quote here part of the question with emphasis that I have added:
While this wastes my time, and that of the other customers waiting, it's not the main concern of my question. Some (again, especially elderly) customers will become furious at me for not understanding their problem, and deny all of my attempts to clarify. This has led to security being called a couple of times already.
My reading of the question is that the "primary focus" is on de-escalating the situation, preventing "furious" customers, and eliminating the need for security to be called on people. That may or may not be what my answer should focus on or it might not be what the asker is actually looking for, but I have decided to focus on that.
As such, the content of my answer is mainly about de-escalation, helping users calm down and feel like they are being given good customer service so they are comfortable becoming part of the solution. My answer is not about how to troubleshoot IT problems directly, since that doesn't seem to be the biggest issue the asker has. I strongly believe, based on my experience on both sides of support calls, that establishing a positive (albeit temporary) relationship on the call is not just beneficial to solving the underlying IT problem, it's critical. If this is starting to sound like I'm suggesting that IT people need a lot of people skills to do their jobs well, then you're understanding me completely. I myself got into IT because I understand computers better than I understand people, so I sympathize with anyone feeling like it's a big ask to see a "computer job" as a "people job". The bad news is that all jobs are people jobs. So you can't escape it, and you might as well accept it and think about people skills.
On apologizing
Some comments (see the chat link) were about apologies, specifically that they are actually detrimental, patronizing, accepting of blame, or merely a waste of time.
First, (and I've added this above), apologies must be sincere. It is true that an insincere apology will come across patronizing, at best. I agree, if anyone is trying to de-escalate a situation and is considering an insincere apology, they should not do it. An insincere apology is patronizing. A sincere apology is a very powerful statement of sympathy, at least in American culture.
In the US, an apology is not an admission of guilt or fault. There have actually been US Supreme Court cases regarding this. Apologizing for something that is your fault merely opens a door to possible forgiveness. It's sincerely apologizing for something that isn't your fault that is a powerful show of sympathy - for the exact reason that it isn't your fault!
On validating illogical or untrue statements
My answer is not about making the world a better place or making the caller or the asker better people. It's about talking callers "off the ledge" and helping them arrive at an emotional place where troubleshooting can begin. If I may quote Obi-Wan Kenobi: "You'll find that many of the truths we cling to depend greatly on our own point of view." By validating another person's "untrue" or "illogical" statements, you are accepting that they have a different point of view and that's ok. I've spent a lot of my life telling people how they are wrong, and I can assure you it's not a way to win friends, influence people, or de-escalate tense support calls.
The tin foil hat crowd who think Bill Gates hates them personally and wants them to lose all of their Word documents might have, maybe, a unique point of view, but they are customers, and if when they aren't, no IT support person is going to ever convince them that they are wrong. The best you can do is get on board their little mental roller coaster for a few minutes and help them out and then get off and be glad most users aren't like them.
On educating callers
My other career is education. I teach guitar and other musical instruments and I tutor in mathematics.
You'll never teach a single fact to an angry caller. Heightened emotions do increase the effectiveness of our memory, but they also filter our perceptions. That means what an angry caller is going to remember very well is how mean you were to them (in their perception). So you have de-escalate if you're going to teach anything.
Effective teaching also requires a connection - a sometimes brief but honest relationship. All of my tips and techniques above are in service of the goal of creating a sincere and effective connection and relationship with the caller. If you think making a connection with an irate caller who needs IT help is hard, try tutoring 13 year olds in Geometry. In both cases, you have to do whatever it takes if you want to actually help.
Finally, the asker has mentioned they are generally short on time and that the callers are frustrated and/or angry. These are not factors conducive to teaching. If the queue is mostly empty and a call comes in from someone who is calm and able to communicate effectively from the first sentence, that's when you can give them a few tips or inside information that will help them in the future.
I highly recommend two books that basically gave me people skills, along with a lot of practice and learning the hard way:
How To Win Friends And Influence People by Dale Carnegie - I know what you're thinking. It sounds like a bunch of cynical manipulation techniques to make you rich. It's not that. This is a great book about dealing with people and I was extremely surprised at how wise it is and how much insight I gained reading it.
The 7 Habits Of Highly Effective People by Stephen Covey - Again, it might sound like a cynical book for salespeople that is trying to capitalize on a trend in self-help. It's not that. This book basically gives you tools and process to become a better person.
Neither book will solve everything, and for some people neither book will solve anything. But they are both worth checking out if you are a computer person trapped in a people person world. Also if you're just a person they might really help.