Can the forward step be part of the modern stroke?
I would say, NO if you mean teaching children and beginner level the modern game. Stepping in on the forehand impedes the player's ability to get the hand close to the ball and find the ball well. Try this experiment:
stand at the net with a friend facing each other. Reach across the net and shake hands, as in an introduction. Now turn 90 degrees to your right (facing the side fence) and shake hands with your friend again. You will find it extremely awkward having to reach around and twist your torso to grasp his hand. Much more comfortable and efficienit to face him.
Forcing children and beginners to step in and get sideways to the ball prevents them from finding the ball optimally. In the modern game power and control come from pulling up and across the ball from the contact point to the finish, not from stepping in and taking the racket back. On the 2hbh it is also more efficient to step with the outside foot to get the non-dominant hand close to the ball and pull up and across to the finish, driving the shot with the non-dominant hand. If the player drives the ball with the dominant hand then he will tend to step into the ball, which gets the hand closer to the ball.
For many players the habit of stepping in was ingrained very early on in ther development and could be difficult to change. On the run, however, the player will resort to hitting open to get the hand closer to the ball for better contact and control. This is usually done intuitively. In my experience when a player is given the opportunity to hit open stance, either from the beginning or as an alternative to stepping forward, they perform better and prefer it once they get the hang of it. I see many, many coaches teaching to step forward exclusively, so their students don't get the chance to experiment.
The concept of stepping in to impart more force and momentum to a shot is pervasive, but very often at the expense of the player's ability to control the ball.
Stepping in on approach shots would, of course, be technically necessary to move forward; employing forward momentum strategically is popular in the current Spanish style, but this is more tactical than mechanical.
On the 1hbh the player must step forward (or more accurately move toward the ball), to get his hand close to the ball, but key to a well-executed 1hbh is the release of the step to finish the stroke.
On modern volleys forcing a forward step impedes the optimal stroke. Again, it is getting the hand close to the ball with proper alignment and moving the racket across in front of the body that produces the best stroke, not stepping or foot positioning.
These references to "modern" tennis are specific to Modern Tennis Methodology, not generalized so-called modern technique. Although MTM does promote teaching all levels to "play like the pros" that does not mean that there is one singular way to hit the various strokes, as the wide varianices of the pros illustrate. On a lesser gradient, however, any player can utilize the basic principles that the pros most often employ in actual match play, regardless of their early training or habits. Like the rest of us, the pros may have certain comfort levels within their stroke production which may be difficult to alter. Even the pros could sometimes stand to modify some mechanical elements for better performance; certainly club level players could benefit from experimenting with different techniques.