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It could be wild onions. They will take some time to get rid of. If you hand removed them, even leaving one little piece will resprout. And using an herbicide takes time too, because the weeds have a waxy coating that is resistant to the herbicide. Here is more:

https://www.gardeningknowhow.com/plant-problems/weeds/killing-wild-onions.htm

They are dormant in the summer and that is the best time to move them. If you are moving and have no other choice, go ahead and dig them up but I suggest planting them in a container, rather than refrigerating the bulbs. And get as much of the rootball as possible.

It depends on how far they broke off! If the tip is broken, but you can see layers of unharmed growth, then it will likely recover.

Plant as you would, normally, but add some wettable sulfur the the wound before planting. This will offer protection against fungi and bacteria.

https://www.gardeningknowhow.com/ornamental/bulbs/crocus/how-to-plant-crocuses-and-proper-crocus-flower-care.htm

Make sure you have saffron crocus bulbs, which bloom in fall. Regular crocus bulbs bloom in spring. Since yours is trying to grow, they should be the right bulbs.

Here is more:

https://www.gardeningknowhow.com/edible/herbs/saffron/growing-saffron-crocus.htm

You will want to keep them uncovered if you want them to survive. Likely, they will not survive being covered for extended periods.

However, if you cannot get the supplies moved, elsewhere, you can dig up a patch to preserve in container until the work is done. Just be sure to get a very large chunk of soil so that you do not disturb the roots of the patch, at all.

https://www.gardeningknowhow.com/ornamental/bulbs/crocus

As long as the bulb is not cut, they should be OK. However, they won't get the benefit of the leaves collecting energy for next year's blooms as it sounds as if the leaves left with the blossom. So, they may be slow to recover. It could be a couple years before you get blooms again.

The mulch will not burn anything. I would point out your bulbs to the lawn service, though, to make sure they don't remove any more foliage or cover the bulbs with mulch.

https://www.gardeningknowhow.com/ornamental/bulbs/crocus/growing-crocus.htm