But if such a process can be done, then there must exist somewhere where I can downloaded a backed up version of a fully synced Bitcoin node, and then just carry on from there, without having to go through the tedious process of downloading and verifying from genesis myself?
But, that's the whole point, man. Think of it.
If you DO NOT verify the blocks, how will you be sure that what you're viewing is your actual balance? By downloading and verifying Bitcoin Core in a clean machine such as a RPi, you ensure that the correct chain is yours once you finish syncing. It wouldn't differ (a lot) if you connected to someone else's node with downloading the blockchain from a third party.
The magic part is the verification.