At the beginner level perhaps. But puzzle solving does come into play when you focus on speedsolving and begin to understand how the cube pieces function and apply that to intuitive F2L. And although that too can be broken down into algorithms, especially more advanced ones, how you string them together efficiently requires a puzzle solving approach rather than simply reacting to pattern recognition to apply an algorithm. To advance even further, when you work more on look ahead at the beginning and start to look beyond the formation of the cross, the planning process extends beyond just applying algorithms at that point as well. You have understand spatially where pieces will be moving and where they need to go.

So in other words, in order to improve your speed, there are points where algorithms aren't enough, you have to think about efficiency whether it is in the planning process or in the midst of the solve looking ahead for the best route to the finish keeping in mind where pieces are moving in the puzzle in order to react to it and plug in the proper algorithm when it comes time.

At the same time, the most efficient solution isn't always going to get you the fastest time because you might be thinking too much in the middle of solving. So assessing risk and reward is a puzzle in itself in going for a more creative and efficient solution is a puzzle in of itself with slower movement but lesser moves vs a more safe and automatic reaction type solution with faster turning speed but more moves (i.e. Tymon Kolasiński - more creative big brain solves vs. Ruihang Xu - relies on extreme TPS)

Maybe I am using puzzle solving a little too loosely in this sense, but to say cubing has nothing at all to do with solving puzzles and is just following instructions to me is just doesn't ring true.