Students who go on to be math majors will get a later course in theoretical calculus. The vast majority of science and technology students (i.e. non math majors) will NEVER have such a class. And not need it, either (to support their mechE, chemistry, etc. majors courses).
So, I think the current approach is fine. The math majors get taken care of with rigor, later. The others don't, but don't need it. This is more efficient than cramming rigor into people who will never need it. And it preserves option flexibility for students not sure if they will study math or science, after high school.
I would also add that it is not pedagogically clear that rigor prior to manipulation is the best way to learn a difficult topic, given the inherent imperfections of "meat computers". For instance, would you try to force first graders to learn formal properties of numbers before arithmetic? Would you make high school algebra students learn Galois theory before solving quadratics or factorable higher polynomials? Do I have to have a perfect handstand (which is non-trivial on rings, try it) before I'm allowed to just swing circles on the apparatus (as a beginning gymnast).
So I think the current approach is just ducky. They even really get SOME exposure to the theory topics, but are not required to master them. Maybe a little bit like how LaPlace transforms are handled within the time restrictions of a typical ODE class. This is done as an exposure, with some basic translation back and forth. However, transform mastery or derivations are really only done by the EEs and systems engineers, who use that topic a lot. And they do that in more specialized classes, later. But for the mechEs, at least they have briefly seen it. So if some minority of them (say doing controls work) needs to dive in more later, they'll at least have sort of heard of it before before diving into more difficult/detailed work.
All that said, I took AP BC in the early 80s at a rather competitive public US high school. So we did see the epsilon delta. And I was fine with it, but I was spending an inordinate amount of time on calc class. But most of the (pretty strong) class hated it and had other demands on their time from tough courses in chem, English, etc. And the calc teacher said, you won't need any of this for the rest of the course when we do partial fractions and related rates and all that jazz. And she was right.
OK, yeah, yeah, evidence by anecdote. But the point really is that very few people will need/benefit from the rigor push, when learning high school calculus. And it is important to think of the overall audience. NOT "well, I could handle it" or "well, I like it", but instead think about the audience and their needs/desires, which may be DIFFERENT from yours.