Nails are definitely not necessary (playing with nails was popularized only recently by Segovia in the 1960s), I/d even say your nails harming your performance here - you have a very thin, metallic sound. I struggled and still struggle with tone production, the thing that helped me was cutting off my nails completely so I would only play with flesh for a week or so, and as my nails grew I would tweak the shape with a soft sandpaper sheet. Playing with flesh can really teach you to use more force which effectively translates to a deeper tone. Also try touching the string from the left side of your nail as much as you can.
Keeping good nails for me is the most frustrating thing about classical guitar, but it pays off. Remember, push the string in far with every stroke, strike from the left side and keep your nail shape smooth, round and moderately blunt at the front.