I did this exact thing by removing the existing system and adding shelves! See the whole process here - https://www.charlestoncrafted.com/creating-a-craft-supply-closet/
It depends on how much you want to spend. I would put in freestanding utility shelving and then put bins with labels on the fronts.
Add a desk and a shelf or two on the wall. You can use two file cabinets and a wooden top for the desk.
What are the dimensions of the closet? Is there an electrical outlet inside?
i have shelves in the top of my 10 foot long bedroom closet. I did not put shelves all the way to the inside corner so I could hang thins, like wreaths or forms. Shelves go half way down the closet. I slid in a shelf that holds canvas bins. In the other corner, I found a wooden shoe bin and laid it on the side to put my mesh in. Clear labeled plastic boxes are on shelves with different things I use, so I can see what I need quickly. More canvas bins are on their side and hold patterns, and such. Fabric resides in a armoire, folded on comic boards. Easy to see! Good luck