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Your tree is a grafted tree, and it seems the tree is only growing from the root stock.
There isn't really a way to know what the root stock is. You may want to look for a replacement.

A couple of tips and an encouragement of patience.

Plant them now in full sun in a location with deep, enriched soil and excellent drainage. The neck of the bulb should be exposed above soil level. Water them while in active growth and after flowering. Then wait — not just one year or even two; by the third spring you should start to see more stems and more flowers.

Do not fertilize these trees. They will fix nitrogen despite having the proper bacterial colonies, although they will likely find replacement colonies just fine. They are very widespread, and if you can grow peas in your area, then you can grow these.

Though a good mix if beneficial microbes will help. These can be found at many gardening centers, or online with the help of Google.

This article will help you with the care of these trees: https://www.gardeningknowhow.com/ornamental/trees/mesquite/growing-mesquite-trees.htm

Without seeing the damage, it will be hard to know for certain. It will be most likely that a couple of the trees could have contracted a fungal infection. This is, absolutely, due to timing of pruning.

These trees should really only be pruned during dormancy, but they can take a very, very light pruning in spring.

I would apply a fungicide to the parts of the tree above ground. This article will help: https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

You can also treat the ground around the tree with DOLOMITIC LIME and WETTABLE SULFUR according to all packaging instructions.

This article will help you to prune these trees: https://www.gardeningknowhow.com/ornamental/trees/mesquite/mesquite-tree-pruning.htm

This one will help you with general care: https://www.gardeningknowhow.com/ornamental/trees/mesquite/growing-mesquite-trees.htm

Since none of the trees have produced pods, it seems that environmental conditions must be the culprit. Our best advice is to make a visit to your local extension service and find out what might be preventing flowering. However, it takes 3 to 5 years to bloom and fruit.

Perhaps this article will shed some light on the problem: https://www.gardeningknowhow.com/ornamental/trees/mesquite/growing-mesquite-trees.htm

It is unfortunate to see watering instructions such as this from a nursery.

It sounds like it has suffered from overwatering. The best indicator that it should be watered, while establishing, will be that it is completely dry down to about 3 inches before watering again. Watering before, especially with these trees, can easily lead to overwatering damage.

You will want to wait until Spring to prune it. Patience and good watering practice will pay off, here. This article will help: https://www.gardeningknowhow.com/ornamental/trees/mesquite/growing-mesquite-trees.htm