Can a key exchange protocol exchange a one time pad instead of a key? As I understand key exchange, two parties can agree on a fixed number of secret bits in the presence of a listening adversary.  That set of bits forms a cryptographic key which is then used to encrypt data using a standard cryptographic algorithm.  And that key is repetitively used over and over, perhaps for an AES session.

What if instead of a key for cryptographic algorithms, a one time key was exchanged for use with a one time pad construct? My system would then exchange another pad for the next message.  Could you realistically securely exchange 1280 bits /160 bytes for a Twitter length message?

Note.  A true random number generator is available.

EDIT.  From initial comments, it seems that the security of the OTP key would be contingent on the security of the key exchange mechanism, say Diffie-Hellman, RSA or similar.  So I guess a fundamental part of my cunning scheme would be the ability to use a key exchange mechanism that provided > (160*8) bits of security.  Wouldn't that then mean  the security proof of the OTP was still maintained?  This is a theoretical exercise, so performance is not a consideration (unless we're talking of geological timescales that make it de facto impossible). Could it be done for example as one message per 24 hrs using a modern desktop machine?  An SOE agent might only send one message per week in WW2.  Just some temporal context.
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.