I don't have what I would call an 'emergency reserve' as such. I have income designated to cover expenses, income designated as disposable and income designated to savings/investing. The disposable income does sit as cash in a savings account and does float up and down. I've never had an 'emergency' it wouldn't cover.
Forummm, the S&P 500 shed 46 percent from August 25, 2000, to October 4, 2002. Just five years later, during the housing bust, the S&P 500 lost 56 percent of its value from October 12, 2007, to March 6, 2009. Now connect that with your statement, "I think even if you did lose your job at the same time the market crashed, it would be better in the long run to have your funds invested. Even if you get unlucky and have to sell it at 50% of peak value, over your lifetime those funds would probably have doubled and redoubled several times."
Where did the 'over your lifetime' come from? Suppose you had FIRED 2 years before the 'emergency' came up and it came up during one of those drops. If you had had $20k sitting in the bank it would still be $20k. If you had to sell on the market, what was $20k would now be $10k. So to get $20k you would need to sell what had been $40k.
An 'emergency' is not a 'long run' issue it is a short term issue. You cannot point to the long term as salvation. Once you've sold, those shares are gone. You can't look to the long term to build their value back up when they are gone.