Myself, I'd approach this by keeping the mesh fixed and unchanging, and using a texture to shave away the edge.
You can make a signed distance field of your nail shape, and use this in your shader to get a smooth, crisp edge on it, even if the texture you're working with is quite low-resolution.
I didn't have a nail mesh handy, so here I'm using this SDF texture on a stretched sphere to get roughly the curvature I wanted. Note how we can get an edge much smoother and more detailed than our underlying triangle mesh, even using just a 128x256 texture.
When you start a fresh nail, you'll blit your source nail SDF texture into a RenderTexture, letting you modify it frame by frame.
By blitting tool shapes into your RenderTexture with "min" blending, you can shave away from the edge of the nail. For example, blending with a linear gradient will give you a straight-line cut-off, like you'd get from a flat nail file. You could also make textures with arched gradients to simulate nail clippers, etc.
This subtractive process is super cheap - similar in cost to drawing a particle - so you can stack up lots of small modifications frame by frame to shave and sculpt the nail in realtime.