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It's hard to tell but is that bermuda or St Augustine? It's being mowed like bermuda so I'm going with that.

You will never ever get bermuda to grow in the shady areas. It will get thinner and thinner until only weeds are growing.

You can plant St Augustine in there and it will grow nicely. When the St Aug grows out and meets the bermuda, then you have issues. If you continue to mow low the St Aug will look like weeds. If you mow it high like St Augustine likes, it will wash through your bermuda and kill it all. Remember that bermuda can't survive in the shade? The coarse St Augustine blades provide plenty enough shade to shade out the bermuda.

Aerating won't help, top soil won't help, fertilizer won't help, and seed won't help. Aerating this time of year invites weeds. Top soil will change your drainage and may cause structural problem in your house. Fertilizer will only fertilize the weeds, since that's all that's there. And new bermuda seed will not germinate until late May or June (it needs hot soil).

How do you feel about removing those trees that are shading the grass?