Not a good idea. Thirsty evergreens like leylandii will cast a lot of shade which will not do your lawn any good at all, and deciduous trees will not draw up any moisture when you most need it in winter, and will result in a lot of moss, followed by bald patches in summer.
You could try aeration with a tine fork, but I'm afraid you have already hit on the real solution, which is to dig it up, find out what the problem is, prepare the soil properly, and returf.