The ISLM model persists in most undergraduate textbooks, and even though it has vanished from graduate courses*, it remains central to the way economists think about policy issues. The main reason it survives is not so much that it allows a relatively simple intro to policy issues to undergrads, as some argue, but the fact that it captures the interaction between real and monetary phenomena which was central for Keynes’ General Theory (still worth reading), and is flexible enough to accommodate divergent views, often reflected in different inclinations of the curves.
It is also true that the conventional interpretation of the ISLM is fraught with problems. But the ISLM model is incredibly flexible and can accommodate most critiques. For example, a few years back a reader asked Mankiw why he still taught that the Fed controls money supply, rather than control the rate of interest, which would be more relevant. His reply was not particularly good (just an ad for his book), but the fact remains that it is easy to change the LM to reflect the fact that central banks control the rate of interest. The ISLM with a horizontal LM is sometimes referred to as the ISMP, where MP stands for monetary policy rule, and for the most part this is done for the benefit of students (i.e. you can buy the new text with the updated ISMP model for $150!).
The ISLM can also accommodate an investment function in which the level of activity, rather than the rate of interest, is central, which is more empirically accurate, and different consumption functions in which other elements besides disposable income appear. More importantly the ISLM does not imply a natural rate of unemployment, which often appears in the supply side part of the macro course. In other words, the ISLM allows for relevant discussion of policy issues, and clarification of policy differences. Besides, compare this model with the consumption theory you get in micro, and this is still one of the most relevant things you can learn in economics. The other would be not to trust economists!
* David Colander suggests that it’s not taught in graduate courses because more sophisticated dynamic stochastic general equilibrium models (DSGE) are more important. DSGE models are based on the aggregation of individual maximizing behavior, and have been less relevant for policy analysis than old fashion macroeconometric models based on the old Keynesian theory. The whole conference on the ISLM, in which you can find Colander’s paper is available here