what you are describing is no longer blockchain and has no cryptography involved anymore. it is just a database like what banks have where you own an account which has an associated balance with it. and unless you can find a way to decentralize it and include cryptography in it so that there is no way of cheating that "balance" without needing a centralized entity to update these balances and control fraud, you won't be able to implement your idea.
like Craig Wright said, the best protection for anonimity is to much data generated like gigabyte blocks.
yes, the best way of remaining anonymous according to Craig Scammer Wright is to always connect to NSA and report all your activities down to every detail willingly

because that is where we land with big blocks, a handful of nodes run by NSA which you have to connect to.
Yes it can be a blockchain, there still will use math and math is math, 2+2=4, its like the block criptography, just will have the total account balance as more a variable to be encrypted in data block and cannot be changed without redo all calculations again, same thing as now in blocks cyphers.
NSA can run nodes too, everybody knows bitcoins its not 100% anonymum, if you want privacy should be better to start to delete first blocks.