Not sure what kind of vehicle you have with the singletree. This site has a Meadowbrook Cart below the red 4-wheeler, halfway down from the top, shows clearly what I expect to see on a nice wooden vehicle. Under a "closeup of the basket" the singletree has a center bolt, goes thru the center, down thru the crossbar, with nut and washers on the bottom side. There is a couple metal fittings between the singletree and crossbar, that work and help swiveling. Shows how the leather restraining straps are usually fitted, not tight or loose. I might have more length on them with a big striding horse, so he doesn't run out of play so quick.
The red 4--wheel vehicle about halfway down, shows the modern singletree, which uses Quick-release snaps on the traces. Not hooks or clips as shown on the Meadowbrook cart for buttonhole trace ends.
I would agree that the straps on the ends are to prevent singletree from swiveling too far, if traces do fall off. Traces can work loose if horse is going down hill with slack or he is pulling cart with the reins, not traces! Especially if holes are worn much, they slip easily. Some folks hook traces with a bungee between holes, to prevent losing them.
I would get the bolt out that is welded in. That wobbling stuff is extra wear, not acceptable to me for using. The more wobble, the more uneven the wear, shortening the life of bolt. Just replace the whole bolt thru everything. Use HARDENED bolts, so they wear better. Most store bolts are soft, wear quickly. We carry a spare bolt with nut, in our spares kit in the vehicle. If bolt breaks or works loose, you can replace it easily, wherever you are.
Maybe your singletree and the bolt, do not have those metal wear parts that stabilize the swivel in everything above the main crossbar. Again, should be no wobble.
Cart or light buggy horse is not having to pull a big load, like logging horses do. Full neck collars spread out the weight of load onto a much larger surface of horse skin. Breastcollar harness is more likely to fit ANY horse at ANY time of the year. Extremely adjustable in fit for fat winter, more fit summer, driving horses. For most fun driving, just tossing on the breast collar harness won't sore up the horse who has not been out for a month, unless there is a dirty spot rubbing against the skin.
Full neck collars need to FIT WELL to do the job correctly. A horse kept in work year round, will probably not need much in collar changes to keep well fitted while working. However for most of our pleasure driving, the horse we use goes thru several body changes over a season. Each change will require a different fit to his collar to make it fit correctly. Adding or removing the collar pads are NOT going to work that well. You will sore him up. He also will get "scalded" from wearing a collar if he is not used to it. You can really "leave some marks" on a horse working him in a collar he is not used to or that doesn't fit him well. Skin needs to also get used to a collar. I have seen horses get scalded in less than 15 minutes!
Few folks want to invest in so many collars, so breast collar harness is a much better choice for their animals, prevents many problems and soring.