nodes could share portions of the blockchain and unspent transactions
Sounds good, would never work. Bitcoin is built to be trustless. Unless you can find an implementation which nodes can store parts of it and still have it verifiable and diversed in terms of its difficulty to execute a DOS attack at a specific node to hinder the access to parts of Blockchain, it will never be possible.
Exchanging portions of UTXO (unspent data set) with Merkle paths serving as proofs for them is perfectly possible, once some kind of UTXO commitment is added to the blocks (e.g. into coinbase tx first).
Once that is done, even miner can download a few last blocks, see their UTXO Merkle roots, then download entire UTXO set from nodes and start mining.
There are like 4 proposals and 3 Python proof-of-concept implementations for that.
After that, rather than growing blockchain we would have to deal with growing UTXO (OP_RETURN transactions, old amounts below dust threshold too expensive to spend in terms of fees, etc.)