Except it's not that simple.  That science stuff gets in the way.  Here's the simple counter-argument to your "simple" analysis:  More access to birth control = much more sex = more failure of birth control = more abortions.
The actual complexity of the analysis is far greater given all of the uncontrolled variables (e.g., acceptance of single-motherhood, government funding of single moms, changing attitudes about sex)  involved in any attempt to analyze the statistics on the subject.