Quite an interesting thought. It has the right to exist, but personally I fundamentally disagree with it. Now I'll try to explain why.
Firstly, there are already some mini-games in the game. Can't remember exactly where you met them, but there are definitely some QTEs in the game (it seems to be somewhere related to cooking and tennis matches). Although I love QTE, imho they look strange in this game, since the player is not ready for them.
Secondly, the implementation of minigames. To make them on the engine for visual novels, you need to spend considerable effort writing code. Besides that, you need a painter to draw it all. Is it reasonable when, for example, not all routes still have backgrounds? Maybe it's better to spend money and time in order to draw more scenes for some special events? Am I wrong?
And thirdly, the game design. It's a visual novel after all. Yes, you can say that you are "playing" it. But it would be more accurate to say that you are "reading" it. This is not some kind of arcade game. It is just text with pictures to improve the perception of the story. It's like a comic, only with choices in storytelling. The addition of mini-games such as tetris does not fit the genre of the game.
I will continue the last point with a fairly clear example. I am currently digging into the 14th day of the Coach's route. With no spoilers, you can play various games there such as: basketball, throwing the ball at pins, ring toss, etc. Whoever played will understand what I am talking about. It is the most suitable place for minigames, right? But I like the way it's done right now more. What at first glance does not have a story branch actually does have. It is based on the choices in the past days, which are not obvious, but in fact, very logical. There are also more understandable types of mini-games. For example, tennis matches. You have the description of the situation and have choices to make. And they determine the progression and result of this "minigame". That's what I call a good game design.
TLDR In my opinion, this game can be even better (and it will be for sure). A lot of things can be added, but mini-games are not exactly what it needs. At least in the main story.
P.S. While I was writing this text, I came up with the idea of how it might look like in my opinion. A separate tab in the menu, with mini-games such as blackjack, chess, 3-in-a-row, etc. against specific characters. Or even more complex minigames with unique designs(from QTE to point-and-click) to fit characters personality(such as: dansing, painting, cooking, fixing etc) And for motivation, if you win, the opponent takes off some of his clothes. Hee-hee-hee. Of course, after completing the main story. We don't want to ruin the first impression of seeing them naked, right?