I use Bitcoin Core, mainly because I didn't realise there was an alternative and I am probably too stupid (and old) to use something like Electrum or Multibit.
Anyway, I hear that the blockchain is now 22GB and growing daily and takes around a full day to download. Why the hell would any new user use Bitcoin Core or bother to download the blockchain if it's not required and the big question:
I dont know about new users, I run full nodes (a bad one at home and a proper one on a VPS) to support the network.
What would happen if every single Bitcoin user switched from downloading the blockchain to software like Multibit?
Bitcoin can not work if every single person is using Multibit. Multibit needs full nodes to work, Miners are full nodes and big companies are advised to run full nodes (yes plural) to be able to judge transaction propagation.
The idea is actually in the original paper by Satoshi [1]
[1]
https://en.bitcoin.it/wiki/Thin_Client_Security#Header-Only_Clients