Wow, that's awesome to hear! One of my fears is having to keep blogging continuously, but luckily math is pretty evergreen, so my intermittent writing schedule doesn't punish me too much.
Hi, I hope you take my comment well. This need to avoid some theoretical logical contradiction far down the road is similar to explaining the perils of split infinitives before teaching a child to say "I want food."
99.9% of calculus students are there to expand their mind. They will never design a quantum mechanical reactor and think "Wow, I'm getting all these weird results, my calculus education must have held me back."
It's generally preferable to give students a first order approximation at first, the successively refine with additional terms down the road. The fact that most calculus students will have no idea I'm referring to a Taylor series explanatory strategy highlights the problems with an overly rigorous introduction. Math at this high level should primarily expand your thinking, and secondarily your storehouse of previously proven rigorous statements.