On the other hand, very brute numerics work for an awful lot of scenarios - that's why epsilon-delta came centuries after Calculus was invented.
For instance, "first-order optics" and "third-order optics" rise from chopping off the Taylor series after the 1st and 3rd term, resp. And it works! In many places, 1st order approximations are just good enough. A lot of scenarios are inherently stable.
So I don't think the intuition you build up is wrong - it just has a scope. There's nearly always a place for counter-examples where "things work the way you think they should" wouldn't apply, however you think about things :)
On a philosophical note, the continuity is a human construct - down there, things seem to be discrete, just with a very small step size. Continuity models these pretty well, until it doesn't - but that doesn't mean the intuition you build up is wrong. Just limited in scope.
On a more serious note, you can understand most, if not all, of Calculus by saying that dx=0.0001, and that A ~= B if they don't differ by more than, say, 0.01 (say, that's the instrument error).
Then you get your limits, FTC, and so on, and verify the results with a four-function calculator.
Example: f=x^2, f' = ?
(f(x+dx) - f(x))/dx = (x^2 + 2x*dx + dx^2 - x^2)/dx = 2x + dx = 2x + 0.0001 ~= 2x
The mental effort you have to make here is that things on the LHS of ~= are "actual" values, and on the RHS are "measured" values, and that ~= is not an equivalence relation.
On a yet more serious note, learning about differential forms will help justify some of that high-school notation.
On a philosophical note, Weierstrass is not the end-all of Calculus. Neither Newton nor Leibniz did it that way. By adding rigor, some argue that the essence has been obscured (hence the non-standard analysis above).