Assume that Discrete logarithms can be solved in linear time over any group (hence factorization is also trivial by a result of Eric Bach), is there any other candidate public key exchange problem that will facilitate computationally secure key exchange?
The Wikipedia page on "Post-quantum cryptography" provides a list of proposals for PKE resistant to quantum attacks. Quantum algorithms can solve DL in finite abelian groups (as well as a few nonabelian and infinite abelian ones), so they get very close to the spirit of the question you are asking. The learning with errors problem mentioned by one of the commenters is on that list.
I do not know of a general answer to the question of "What is the power of DL in arbitrary (that is, nonabelian) groups?".
Check Supersingular Isogeny Key Exchnage for some nice work on a Diffie-Hellman like Key Exchange based on isogenies between supersingular elliptic curves. This will compute computationally secure key. The authors mention that even though the key exchange is secure it will most likely be interesting purely for research/ pedagogical reasons.
There also is a cython implementation by De Feo SSIKE git. I would not rely on that for cryptographically secure key exchange but it is a good starting point.
Additionally to the wiki page mentioned Adam Smith you might wonna checkout this website for a good overview of literature: pqcrypto. On that page you will also find a link to this document: Post-Quantum Crypto Introduction. Thats the introduction to a book called ''Post Quantum Cryptography''. On pages 6-10 you will find a technical overview of three ''common'' post-quantum cryptosystems.