I think you're <i>describing</i> <a href="http://books.google.com.sg/books/about/Algebraic_Topology.html?id=h-wc3TnZMCcC&redir_esc=y">Spanier</a>.

Everyone I know who has seriously studied from Spanier swears by it- it's an absolute classic. The approach is exactly as you describe- algebraic topology for grown-ups. The treatment of homological algebra in it is extremely nice, and quite sophisticated.