http://www.science.smith.edu/~callahan/intromine.html is one idea (and read that page/book for a critique of the kind of typical ~200 year old textbook/course we still use today), but this could be a lot better with a bigger budget and more support.
It largely dropped the "memorize this set of symbol-pushing rules then apply them to a long list of exercises" version of differential/integral calculus found in typical introductory textbooks, in favor of a "make up a model for a situation, then program a simulation into a computer and see what happens" approach. That makes for a radically different experience for students.
It’s hard to simply say this is “better”: it depends what skills and content you are trying to teach. The more computing-heavy version arguably does a lot better job quickly preparing students to engage with scientific research literature (because differential equations are a fundamental part of the language of science). But it might make it harder for students to e.g. dive into a traditional electrodynamics course intended for future physicists, full of gnarly integrals to solve.
Most of the people proposing even more significant departures (in content or style) aren’t writing introductory undergrad textbooks.
Using simple computer simulations to teach introductory math courses is definitely a change that has been slowly happening over the past couple of decades.
However, different approaches don't just teach different "skills and content" as you say, but entire paradigms of thinking. There is mathematical thinking and there is computational thinking (and other types as well), and any course helps you step up the ladders of these paradigms by different amounts.
My experience teaching undergrad math/physics/cs for several years is that computational thinking is in the short term time and effort cheap, and this causes a fixed point in how students think. If you give them the concept of say differential equations, and teach them some computational methods and some mathematical methods to solve these equations, they will always lean towards just using the computational methods. This seems all fine and dandy, except when you go to more advanced mathematical abstractions, and in the previous step the students had not mastered the mathematical way of thinking, they are lost. They simply don't have the mathematical capacity to grasp the higher abstractions. And no amount of 3B1B fixes it - this lack of long term investment into an important thinking paradigm.
Just look what you can do with high-production-value video animations: https://www.3blue1brown.com/lessons/essence-of-calculus