I am not sure entirely what you mean by “approach” but here are some thoughts.
Art, just like writing, is a form of communication, and your game screen is going to be one of the first things players see, and the most seen thing, so you have to think about, not only what types of players are going to player your game, but how those players will see the game for the first time, and how they will think about it after they learn more of the game.
For example, you make a game, and the focus is on a character and on the world they live, and the world you have to help them through. So say a platformer game in a distopian future that is made with pixel art. In this case one could say that the focus of the game is about the story that the player will experience. If it is going to be about a story, it is safe to say you want to glorify this, so you make an epic art thing to act as the main menu, such as the character looking over the city. Buttons are off to the side, because the focus is the art, hidden in the art is the summary of the story that the whole game will cover, maybe you can see some key locations. In this case, of a story, and you know you advertized your game as having a story, safe to say that your audience who come to play your game are coming for that reason, so they are more willing to work to play your game. So that means it is safe to not make the buttons big and obvious, as well as it is ok to have quite a few, things like continue, new, load, options, extra, quite, credits. Maybe even have a scroling message on the bottom. This may be sensory overload, but since you know your audience have come here for the story you know they are willing to read a bit, so you don’t have to worry as much about scaring players away.
Say your game is simple, very simple, like say all you do is swipe to cut up clothes or something. The could be no menu screen, or it could be a few buttons, but instead of a background art, the game is the background, waiting to be played once you press play, and then the menu slides out of the way to let you play, or the background could be the game, and the game is being automatically played like a little preview before the players play.
Now say your game is designed in a certain way, either focusing on game mechanics or an art style, such as my favorite, 2.5D games. There may not be a strong story or character writing, heck maybe even the game mechanics won’t save you, maybe all you do is walk in the game completing mission after mission, In that case, I treat the menu like it is its own game level. I reuse game assets, some game code, and I do a hybrid of being artistic like a said in the first example, and minimal with a preview of the game in the background in the second example. Since the story isn’t the highlight, maybe just the art, I have to worry about player’s short attention span, so the menu must be very minimal (as a programmer, you can have it say start new game, but also search if there is already a save, so that next time the game runs, it will say continue instead, with only a few other buttons on the main menu to reduce information overload), while at the same time interest players without decieving them, so the art has to be in the same style/spirit of the game, hence why one reuses the art assets as if you were making another level. The menu button is the focus but they do not have to be in teh center, they can share space with some art. The art, and background, must not compete with the buttons for the players attention, as a way of reducing information overload, so you don’t want a complex painting of a world. instead it may be better to keep the background a simple color, maybe some patterns, partical effects, but nothing to showy, that is because in order to keep yourself under control as an artist, you will restrain yourself by chosing to have one particular focus point be in the art (sometimes it is the title, but this shouldn’t be too fancy in this third case, but go ham with making a complex title font in the first example above), a good example would be to the side of the buttons, a preview of the character, possible doing an attact animation, or the art cycles showing off different important characters, or objects, or locations (doing a slide show may work, but it shouldn’t take up the whole background otherwise it looks too basic, to childish and unprofessional), litterally just using sprites and images you would use in game, reusing them and moving them around for the main focus point on the menue will be all you need.
those are some basic cases you can consider and see what you could do.
