To each his own, but epsilon-delta is my go-to example of formalizing an intuitive concept ("gets closer and closer"), which is a high-level mathematical skill.

The intuition and the formalism are presented together (at least, they should be!). To learn the role of epsilon and delta, the student needs to jump back and forth, finding the correspondences between equations and the motivation. This is a skill that needs practice; this was one of the first places I found the equations dense enough that I couldn't just "swallow them whole".

(The earlier I remember is the quadratic formula, which I first painfully memorized as technical trivia. It took me a couple of years to grasp that it was completing-the-square in general form. Switching between the general and the specific is another skill that you develop)

Surreal analysis is sort of a thing but it is quite far out there (e.g. you can have transfinite series instead of merely infinite ones). Maybe you meant nonstandard analysis (NSA), which is real analysis done with infinitesimals, but the machinery justifying it is way outside of what you'd see in even a theory-oriented intro calculus class. There was an intro calculus text (Keisler, 1976) that used infinitesimals and NSA. I don't know how it dealt with constructing them though.

https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...

It’s very enlightening. Math was formalized for a reason. Now you don’t have to argue what something was supposed to mean and whether it is a correct or a false interpretation, eg: https://hsm.stackexchange.com/questions/6867/what-is-the-cor...

Rather than viewing delta-epsilon and infinitesimals as opposing methods, they can be seen as complementary tools I think