Are there any plans in the core development to at least allow users to run the core software without having to download and process the entire blockchain?
This would be great and I would be happy to support this!
That would not make sense. Bitcoin core is as it has always been a full node and as a full node it needs to download and
verify the entire blockchain; you can also decide whether you want to store the entire blockchain and be a full verifying node or store a small number of latest blocks and be a pruned full verifying node.
Additionally there is no need for bitcoin core to add such a feature since we already have an excellent implementation for an SPV client (where you don't download and verify the entire blockchain) called Electrum.