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This is pretty easily done. I like to set mine into well draining soil. Water it until it runs out of the bottom. Don't let it sit in water, but let it dry thoroughly before watering again. It will root on its own without much help. If you would like to help it along, you can crush up an aspirin tab and dissolve it into warm water. Let this cool and use it to water. This can help much faster for rooting.

This article will give you some more information: https://www.gardeningknowhow.com/houseplants/aloe-vera/aloe-plant-propagation.htm

Can you provide photos? This can help me to see what is going on. Are there any animals around that could have damaged them?

Aloe leaves store very well. They will not lose anything except moisture. This will actually concentrate the chemicals within the plant. It would not hurt to buy cut leaves.

If the aloe plant still has some roots, it will probably be able to survive long enough to grow a new root on its own. Just be sure not to overwater it while it's in a damaged state. If the plant has completely snapped off from the ground, you may be able to propagate it using these methods:
https://www.gardeningknowhow.com/houseplants/aloe-vera/aloe-plant-propagation.htm
https://gardening.stackexchange.com/questions/1566/can-i-save-my-damaged-aloe-vera-plant

Planting the Aloe in deeper will likely cause rot in the stems, I would not advise.

It is difficult to propagate an Aloe do to the high moisture content in the leaves. Removing the pups as you did is the way to propagate the plant.

You may want to keep it happy and harvest the remaining leaves as needed.
Aloe Vera are quite inexpensive and it may be time to treat yourself with a new plant.

I would not disturb it anymore. Just make sure you let it dry out very thoroughly before watering again. They are very drought tolerant, and will enjoy being dry for a little while between watering.

As a side note... Your aloe is flowering! This is a rare sight, and something very interesting to see.