I have come across it occasionally, always used in that kind of situation (that is, '..his friends were very instant with him..'), and assumed, without checking up, that it had lost its battle with 'insistent'. I feel, though, that 'instant' implied an urgent insistence, usually pressing for some action before it's too late. A professional advisor may be insistent, but perhaps only a friend could get away with being 'instant'. So I think the usage had a value and rather regret its passing.