Through repeated DH attempts, could Alice and Bob build a large random key for use as a one-time pad? My intuition is that this protocol would be as hard as breaking DH (discrete logarithm).
Through repeated DH attempts, could Alice and Bob build a large random key for use as a one-time pad?
No, since that is not how a one time pad works. You can use your idea to create cryptographic material to encrypt plaintext using XOR, what you are describing is an asymmetric stream cipher of some sorts.
A one time pad requires an "offline" exchange of key material. A one time pad when used properly is unbreakable, whereas a DH exchange can be broken or potentially maliciously interfered with. Using this idea may potentially expose a weakness in the PRNG used to generate the private keys, but that same problem is applicable to whatever is generating a true OTP. I this case however, the public key exchange may expose this weakness much earlier than an offline key exchange, as no ciphertext would be needed.
This idea may be significantly faster than an offline key exchange, but that depends on how much key material needs to be created. In your example, the shared secret would generally be passed through a hash function to distribute entropy more evenly over the bits, or possibly reduce the size of the shared secret so the entropy per bit is higher.