This is exactly what small claims is designed for.
You write the claim, give the evidence, and if they don't show up, you win by default and can hand the judgement to the insurance company to get paid.
Small claims courts typically have fairly low limits, much lower than what the insurance might pay for damages to a house or furniture and art (as described in the write up).
You might get the cost of the move restored, but that doesn't cover the cost of repairs, replacements, and restoration.
You have to serve their registered agent in a way that satisfies a judge. I tried to sue a company whose registered agent was a high profile attorney. I paid multiple people to serve him, but every time it got back to the judge his office had somehow tricked the service officer into accepting the summons in a way the judge deemed invalid. I had to give up because service fees were approaching a significant chunk of the claim and you can’t recoup that in OK…
"I was only ruined but twice in my life — once when I lost a lawsuit and when I won one." — Voltaire
It's not clear to me that a default judgement in a civil case against a moving company would be something their shipping insurer would pay out.
As is demonstrated in this blog post, the insurer paid out directly on a claim of damages, no civil court required. But above & beyond that, to recoup costs paid for the move based on a dispute about the service rendered, my understanding is that you'd end up with a judgement against the moving company and the same struggles chasing them down as you already had.
Why would the insurance company pay out? They insured your belongings during transit, and already paid for the damage. Whether the services themselves were rendered to your satisfaction or not is between you and the moving company. And good luck collecting from them.
But if they do show up and know how to play the game, buckle up. You’ll be in and out of there for years. The judge will ask you to settle for a fraction of what you’re owed. When you finally do get the judgement, they won’t pay it. Then you need to drag them back to court to get them on a time-payment plan… then they make a few payments, stop paying and move to another state. Then you need to domesticate your judgement in that state, and maybe you can start to think about garnishing wages or drafting from accounts of theirs. By then, you’ve wasted so much time and money, you’re doing it mostly for spite at this point.