It is not necessarily a root pruning service - the air spade is just a means of excavating soil to investigate if there even are girdling roots to prune. It is entirely possible that are not girdling roots, and they'll run their air spade for 15 minutes to realize they need not hardly do anything, but better expose the root buttress. The root flare is already visible, which is at least half the battle with the air spade. The air spade can be a very useful tool in the event the tree was planted too deeply, which does not appear to be the case on this tree. For these reasons I would not recommend this service, although I do agree it would be of some benefit, but nowhere near enough to justify the cost. 

If you're comfortable yourself to do some investigative work, you need only cut the turf away from one side of the tree, and proceed to pull back the soil around the whole perimeter of the trunk - a pick mattock is the perfect tool to do this. If you encounter any circling roots, simply sever them. From the second picture, it appears one girdling root around the trunk was removed, which if true, is really good to remove. But such a root, although problematic, would not kill the tree by itself. I would venture a guess, judging by the excessive amount of lichen that the tree is severely struggling, and the heavy soil, likely poorly drained, is the main contributing factor.

And in the case the air spade does have to excavate a foot down or so, let's just say (even though you need not go below the flare), and they remove a many circling roots, and even some girdling, further down the side of the tree without the buttress, the tree isn't going to magically spring to life, it is going to continue to struggle. Even if they were to find a massive girdling root that was severely choking the tree, which they could, and perfectly remove it, the immediate effects would be die-back before a long, long recovery. And in the case you're not happy with how it looks currently, your best bet is removing it and replacing it, try swamp white oak or london plane to better suit these heavy soils, and always make sure the root flare is visible and clean.

This is my opinion based on the two photos. Best of luck.

https://marylandgrows.umd.edu/2024/01/12/free-the-flare-maintain-visible-root-flare-for-tree-health/