you're kind of coming at this backwards. it's not that someone doing analysis doesn't care about whether all functions on reals is continuous, it's that if you hand them a foundation where that's true, they'll disagree with whether your foundation is correctly modeling functions/real numbers.

At which point we would have an interesting debate! I could tell them all about how this foundation will give them a more nuanced view on continuity!

I suggest you go meet some PhD mathematicians and have that discussion.

Having a PhD in mathematics myself, I have been surrounded by such and had this discussion a few times. Some even like the ideas suggested!

I would say the most common counter argument is cultural: Classical mathematics is the norm in the field, hence one must use it to participate in research in this field.

But that seems to me a rather intellectually unsatisfying argument, if one cares about the meaning of the work.