I've never understood this mentality. I've got a credit card with nearly 6 months of expenses as a limit, most of it not used because I pay it off month to month.
I'd be more worried about the tax implications of selling ETFs and such than the 2-3 days of wait to transfer.
Maybe I lack imagination, but I can't think of anything that I'd need my entire emergency fund immediately. Throw the emergency cost onto a credit card, and transfer the money from the emergency fund to pay off the bill. I guess that doesn't work in some other countries where credit cards aren't as prolific as the US.