How I like to think about it is that given an expression with a derivative dy/dx, we can always insert an arbitrary variable s that varies with both x and y, so that we can obtain an ordinary quotient (dy/ds)/(dx/ds) by the chain rule, and manipulate it normally with no qualms about what it means. As you say, second (and higher) derivatives can be calculated with the quotient rule.
What I did in my book to keep everything algebraic but not introduce weird notation is just set the derivative equal to a variable. So, say m = dy/dx. Then, the second derivative is just dm/dx.
The advantage to the revised notation is that you can describe things that are difficult or impossible to describe in the other notation. For example, you can legitimately look at d^2y/d^2x (note the placement of the 2 on the denominator to see how this is different). This is a valid ratio under my system but invalid under the standard system (though I actually consider my system to be the standard system just with prior mistakes corrected).
