Earlier quoted context omitted.
Nice to see that there's no limit where I live which is Norway. Hopefully, there are no plans to introduce one.
I am confused by your post. What are some good reasons to allow cash payments above 3000 EUR?
1) Disallowing a third-party in every single transaction you make, especially as
1a) Inflation means 3k now might be quite low in 20 years
1b) Banks themselves have been shown to be scummy
1c) Banks are a rare institution to allow outsiders into.
2) Bank accounts themselves are not actually all that accessible; it's quite common for people to be ineligible for a bank account under random (but common) circumstances. -- an interesting loop you can get into is that you need to pay a utility bill in order to get a bank account, yet you can't pay a utility bill without a bank account.
3) Cash is the only fiat currency that actually exists, I know it's trendy to think that lines in a database are meaningful- but if you have physical cash you know what it's worth and it can't be hacked or seized. (though a good counterpoint is of course that you can be robbed or it can get destroyed).
I live in Sweden and we're very far down the path of "cash is dead" - there's a considerable number of stores and fast food joints that do not even accept cash. However, when I moved to this country I couldn't open a bank account and without people accepting cash at that time I would have paid exorbitant fees on each transaction.
So with that in mind I'm not in favour of restricting cash, I'm aware of the drawbacks of large fiat currency transactions; but It gives the central banks even more power than they currently have (which is a lot, which you'd know if you were on the wrong side of a policy or get into one of their loops) and we're not sure that they are going to continue to be fair into the future.
Note: you cannot hold a bank account with the central bank that actually issues currency. You are voting for a middle-man.