How do you define real numbers without uncountable sets though? I'm not sure most people are willing to give up the real numbers just because they are a bit unintuitive.
I believe the mantra is "π is a rule, not a thing".
It is pretty standard device in mathematics - having a set and an operation, to produce and explore another set built as a closure of the first set under the operation. Seems pretty intuitive to me :)
What is really unintuitive, puzzling is the starting point of it all - natural numbers... 1, 2, 3... why there is a one stone, a one star, ... why this discretization?
{} == 0
{{}} == 1
...
I'd argue that, for most people's tastes, computable analysis is wonkier than real analysis, even if the latter has uncountable infinities (which is rarely considered a problem).