Can a key exchange protocol exchange a one time pad instead of a key?
In theory it could work, if you can ensure that the resulting key is actually generated uniformly from the full range, as long as one party chooses its input uniformly.

A rough sketch for a proof: Consider the inputs $x$ and $y$ to the key exchange $f(x,y)$. Then if you consider one input fixed the key exchange has to be a bijection. You need to proof that for every $x$, $f$ is a bijection between $y$ and $f(x,y)$.

But on the practical side, I can't think of any realization of this, where we can actually prove that. The obvious candidate is the DH key exchange. But if we use a generator $g$ for the full group, then we can't prove the above property: Assuming any fixed $x$, then $g^{xy}$ is not always a bijection, there is the counter example $x = 2$. If $g$ is a generator of a prime order group, then this property would hold. However, in this case you can only reach the elements of this particular subgroup and not all group elements. You would need a bijective mapping from the subgroup with $k$ elements to $\{0,1,\dots,k-1\}$ or similar. And that should be difficult, unless you assume you are able to calculate the DLOG. And this problem exists for both classical and EC-based Diffie-Hellman key exchange.