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It's a good point about 'credit' for work done. That's a big change for lots of people (extra context: the team in this case is offshore, which complicates things a little in this regard). I don't agree that the planning element of estimation should be considered an afterthought though. Yes, owning and understanding commitment is crucial but the purpose of that commitment is partly to know how much value is added over time and the reason you need to know that is so you can plan around it as a business.
You probably will, yes. But you can avoid this by having a status that isn't mapped to a column. I usually don't map 'Closed' to a column so once I've closed them they are no longer visible on the board. They'll still be findable in JIRA if you need them later. Go to Board > Configure > Columns to change the mappings.