The answer to your question is YES
Max effort does both. the best way to build strength is with high weight low reps. So how much higher weight and lower rep is there than one of the most you can do.
When you hit the weight you can not move, that is where you stop training for strength (as much) and start learning your weakness. Where did you fail on that last rep.
In the past I have been guilty of hitting failure and just surmising that I need to improve that lift without analysising it. I think this is what most people do. But, if you analysis it you can find that your tris gave out toward the top or your lower back is weak. Then you can hit not only the exercise but also that weak area, thus improving your lift faster than if you just performed the lift everyday.
That said you are probably right about elite lifters. I'm not one and don't know any so this is my thought not experience. These guys have a pretty good idea what they can lift on any day and how quickly they can expect to improve. Plus it would be incredibly hard on their bodies after years and years of max effort.
I'm going to assume that you, like me, are not an elite lifter. So in the short term I would say, hit those max lifts and use them as training and learning opportunities.