Post-Quantum Primitives' Object Sizes
TL;DR

Don't look for a primitive that provides the best communication efficiency. Instead, look for ones that are rich with desirable features. 

Asker's confession

As the OP, I've always had a compulsive frustration that we don't get to enjoy the `perfection' and elegance that elliptic curve primitives used to provide in pre-quantum public-key cryptography. ECC operations are efficient compared to RSA; have actual exponential keysize-security relation; and quite a few other nice properties. 

However, ECC isn't always a silver bullet. It requires generating system parameters in complicated ways; and certain parameters must be verifiably secure and sound - which isn't easy to achieve. 

Symmetric-key analogy

It should be noted, that constructing secure and efficient primitives in symmetric-key cryptography, is no less noble an art than that in public-key cryptography.

Some blockciphers may require larger and more complex key schedule; some hash functions may require larger internal states, or more pre-computed constants, or both; some AEAD construct may run fast on some systems, but very slow on others. 

Why answer my own question

The way the question requests an answer can't be say isn't "too broad" - it really is a bit too broad because I think it's quite deep. 

I'd like to answer myself in a way that can inspire thinking instead of pointing people to a mis-predicted future. 

What we really should look for


Communication / object size efficiency. 
Computation efficiency / performance. 
Interoperability. 
Ability to be implemented correctly. 
Ability to be implemented cost-efficient. 


etc. 

All these factors must be balanced.