Why does RSA need two prime numbers?
"I suspect modular exponentiation is much cheaper computationally that regular exponentiation, and that perhaps that is why two primes are used in practice."


Yes, modular exponentiation is much faster.

How much do you think a number expands if you perform non-modular exponentiation? 

And what's keeping an adversary from just trying all possible inputs, and encrypt them to see which one matches?

You can of course random pad $M$ with e.g. 128 random bits, but then your input will grow even larger, the exponentiation slower and the result even larger.



But most importantly, what's hard about calculating $\sqrt[e]C=\sqrt[e]{M^e}=M$ if $e$ is known? I don't think that that's a hard problem even if $e$ is very large. For a large $e$  the calculation really becomes problematic because the result size would be humongous anyway.

Of course, your encryption function needs to be irreversible without the private key value - in this case $d$ - and it isn't.