BU's fundamental purpose is Semi-Unrestricted block building (accelerates network centralization 1).
CORE's fundamental purpose is Semi-Restricted block building (preserves network decentralization 2).
1 Bigger blocks tend toward network centralisation,
3 but decentralises the user base (more people can afford to send bitcoin).
2 Small blocks allow greater network decentralisation,
4 but centralises the user base (only a few big actors can afford to send bitcoin).
your putting words into people mouths.
1. Bu/bigblockers dont want one brand running anything... most bu/bigblockers are happy with bitcoinj, xt, classic, btcd, etc all running on the same level playing field all using real consensus to come to agreement.. and if core got rid of blockstream corporation. would be happy with core too.(main gripe is blockstreams centralist control)
2. small blockers have shown distaste for anything not blockstream inspired/funded.. (rekt campains agains bitcoinj, xt, classic and bu)
3. correct(people keep funds on thier personal privkeys and use future LN services voluntarily)
4. correct(people move funds to new keys and multisigs where payments need an LN counter party signing. but done forcefully due to politics/fee war games)