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Does it get full sun? Usually that happens when a plant is reaching for light. If it is indoors try moving it near a sunnier window. Before it falls over, you can cut it in half and propagate the top part.

Review these cultural conditions and see if they are being met:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/caring-for-and-landscaping-with-yuccas-outdoors.htm

https://www.gardeningknowhow.com/ornamental/foliage/yucca/yucca-plants-care-and-pruning-tips-for-pruning-a-yucca.htm

Starting one from cutting is pretty simple. Just make sure that the plant is not in flower when you do so. Flowering hormones inhibit rooting hormones.

Here is an article that will help:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/propagation-of-yucca-plant.htm

It appears to be normal discoloration. Just make sure not to cut any leaves off unless they are almost completely dead.

Here is an article that will help you to care for this plant:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/growing-yucca-indoors.htm

In your area, I suspect that the soil remains too moist for too long to let them flower properly.

You can prune out old branches, but you will need to treat the soil with fungicides frequently in order to prevent wet soils from killing off your plant.

Here are some articles that will help:

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

https://www.gardeningknowhow.com/special/organic/biofungicide-information.htm

https://www.gardeningknowhow.com/garden-how-to/info/homemade-fungicide.htm

https://www.gardeningknowhow.com/ornamental/foliage/yucca/caring-for-and-landscaping-with-yuccas-outdoors.htm

What you are seeing is a replacement branch, and they will commonly shoot more than one if they are healthy to begin with.

I am noticing signs of disease that have been inside of the small tree for quite some time, long before toddler damage. This is likely why it is not recovering, as they are normally very fast to sprout replacement shoots.

These trees prefer dry soils, and will contract disease when overwatered. Make sure that the tree dries out down to about halfway the container before watering again, and treat with a fungicide to rid it of the infection that it has.

Here are some articles that will help:

https://www.gardeningknowhow.com/ornamental/foliage/yucca/growing-yucca-indoors.htm

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

It does not look like a yucca, which has erect stems. They don't usually cascade like this Is it possibly a ponytail palm?

Black or brown tips is usually a watering issue. Here is more:

https://www.gardeningknowhow.com/plant-problems/environmental/what-causes-brown-edges-on-leaves-of-plant.htm