Defending a diversity of bugs... Relevancy of arthropod biodiversity?

Well... Sam, now, I don't have a great answer for this and would probably flounder. Especially when the first default so many people have is to scream and squash bugs.

I guess... well, I guess with the right person, say you're looking at a boring old moth, well, I guess you could always show them how the moth is actually quite fashionable and wearing some very beautiful clothes, like a Victorian lamplighter shawl, and how cool is that? Here's one that has a green furry cloak; here's one that is dressed in an orange, purple, and green winter coat - do you think the fuzziness helps it stay warm? Look at this one, surely the extra white curl is like a wig!

Maybe that little bit of anthropomorphizing could be the key to some people's hearts. (Okay, probably younger people. I actually tried it with a relative who was very upset about assassin bug nymphs, which the person thought was milkweed bugs, and I got to talking about the business suits they were wearing with cute little black buttons and bowties and honestly, it did not work, I managed to collect the assassin bugs and carry them off before the yard spray came out. I think... we gotta start with folks who are young and have less preconceived notions about bugs being bad.)

First, I share your bias. I love bugs! Second, we shouldn't need to justify bugs or any other aspect of nature. Nature just IS, and that should be enough. But for some, I guess, it isn't.

As already mentioned, there is the "beauty" argument. It's not just moths and butterflies that are beautiful to the eye. There are some gorgeous beetles, odonates, true bugs, mantids, and others. We also have to remember that the beauty that we see did not evolve for us but for some function in the insect's life cycle.
Then there's the pollinator argument. Without pollinators, which occur in so many groups of insects, not just bees. Except for the grains, essentially all of our fruits and vegetables rely on insect pollinators to reproduce. Interestingly, we in North America have become almost entirely dependent upon a single introduced species, the Western Honey Bee, for our agricultural pollination, a strategy that has become questionable with the significant declines in honey bee populations.
There is the "sanitation" argument, as mentioned by @nature-tracker. Without insects, we would literally be up to our eyeballs in s***. Think Australia after the introduction of cattle and before the introduction of dung beetles.
Most importantly, however, is the importance of insects in every food web. When "justifying" bugs to "non-believers," you may have to defend those insects that truly "bug" us, like mosquitoes. As annoying as they are to us, we have to remember that nature does not evolve to serve humans, and mosquitoes have an important role in aquatic ecosystems. Without mosquito larvae, those bass fishers love to catch probably wouldn't be around, either.

I thought about this a little more from a practical angle today after finding a dying owlfly in a bowl of water left out overnight. (It's not my water bowl, but I suspect most people would see it, say "oh, ick" and dump it out without thinking about it much more).

While I really do believe that my silly anthropomorphizing of little creatures that may otherwise be seen as drab or scary can be helpful with the right audience - to make them more 'fun' and relatable - in truth I know I can't expect to pull that out of my hat out with most adults.

Perhaps when we look at "bugs" from such a wide lens it is hard to share the importance of them. We are trying to carry too much weight on broad concepts that may not be so meaningful to somebody no matter how we phrase those concepts. Maybe we need to narrow the focus so that a person can understand the value of one or two things at a time.

I've had some success getting people to see paper wasps differently when I talk about how they keep pest caterpillars out of my garden. I pull up some of the pictures I've taken of them in action, grabbing caterpillars from my leafy greens. I talk about how the queen pollinated my early flowers. I stand by their nests to show they aren't interested in attacking me if I'm just hanging around.

Via another analogy - Sam, I went on a nature walk you did last year at Native Plants and Prairies Day. I don't remember everything but the background history you gave on the Arkansas yucca has stuck with me; I imagine it stuck with a few people, who now might see a space and think of that plant and see the value of the space differently. The people who held pieces of the plant could imagine making ropes or baskets with it. What you did not do was tell the group about the plant's value to the ecosystem scientifically, or speak about it in broad ecological terms - it was story showing the plant's relevance in our society. Bugs -- one at a time, perhaps.

So true.

This is something I grapple with a lot - not justifying my love for bugs, exactly, but dealing with the burden of living in a world where bugs are so eagerly despised. Knowing that in most cases, it's not even on an individual scale - it's a reflection of the bigger picture, a society where there's profit in teaching people to fear and to kill our native wildlife. Sometimes it feels hopeless when I look up an interesting bug or a rodent to learn more about it and the first ten search results are pest control companies telling you what poisons to buy to kill them. Most recently, it was springtails - the article said that although they cause no damage to people or property, they're a pest "due to their large numbers". Why does simply existing near people make them a problem? They're eating the mold growing in basements and in between walls; they're doing us a service! And people will thoughtlessly pay someone money to get rid of them. Sometimes, it seems hopeless to even bother to change peoples' minds.

But we wouldn't be who we are if we gave up the good fight - I think bugs are a good teacher, in that regard. They're eaten by everything, one wrong step away from being crushed or splattered onto a windshield, but still they keep going. You can pluck a leg off a bug and it might not even notice. I owe it to nature to be bug-like in that way; to do whatever I can to share the gifts nature has given me with others. If my enthusiasm rubs off on even one person, who knows what kind of a difference that could make?

I wish the answer to your question was as simple as "You should respect life in all its forms, not just when it's convenient for you." That's the real answer to the question, but it's not the answer that's going to change the mind of someone who doesn't already understand that.

I think you may be more equipped than I am to give the answers that might actually sway people, and the common ones have all already been said (food for other animals, pollinating our crops, decomposing our waste), but here's a few more for the arsenal:

-Removing arthropods from the food web would simply cause it to collapse; if you work your way down, pretty much everything relies on them. And even the large herbivores need them pollinate the plants they're eating.

-They're food for us, too. Lots of cultures traditionally eat insects; even in Western culture, we use shellac and carmine. And if that still doesn't do it: what about the chickens that they eat? Insect protein is a great chicken feed. Related to that, it's likely that early humans were able to grow our giant brains by eating bugs. Eventually, that expanded to include hunting bigger and bigger animals - but insects were probably the first protein-rich foods that allowed us to become the weird apes we are today.

-Silk and honey! Without insects, we wouldn't have honey in our tea or silken pajamas. The silk moths in captivity are completely domesticated - they can't survive without humans, we're intrinsically linked. Honeybees could choose to leave their hives, but the protection offered by the humans who tend to them is worth losing some of their honey.

Sorry I waxed poetic about this, I just have a lot of feelings about the value of bugs and nature in general.