I suppose this could really depend on the part of the world you're in, but there are still many instances of "emergencies" that need "cash". You have to decide how much cash is the right amount, but I still recommend having $1,000 or so in liquid cash. ItIt can really make a huge difference. LetLet me give a few examples.
Bad weather. I live in Florida and after a hurricane if you want gas, food, or water, you're going to need cash. ATM networks are the last to get repaired. Same for internet. Cell services are not at the top of the list. You could be 4-5 days without access to your accounts. If you need anything in that time frame it's cash or the Red Cross.
Children. As a foster parent we keep some cash on hand because as kids come in to our care we want to get them in school right away. How are they going to eat? You can't give them your plastic, you haven't had time to setup "lunch accounts" or the like. Cash always works. Along those same lines are bus tickets, clothes, supplies and what not. Not everywhere will take an ATM card, and if your money is tied up in a stock then what are you supposed to do for 3 days.? Tell your new kid that he can't eat?
Big Emergency problems. Like your car breaks down on the interstate. Sure it can get towed to the dealership, and yes they will add the tow to your bill, and sure you have a few days to pay it (while your car gets fixed), but how are you going to get around? Not all taxis take plastic around here. Almost none of the busses take plastic even though they are supposed to.
Lost Wallet or ID theft. LooseLose your wallet, good luck getting into your bank accounts. It can take weeks to establish your ID after you looselose it in some cases. You want your ID, but you need your birth certificate. You have to GO to the state of birth, and request it. Oh but wait, to travel you need an ID. No problem send away for it, but wait, you need to send them money to mail it. To bad all your money is tied up.
Running or Evacuating - So you need to evacuate for some reason. No big deal. But all your money is in a place you can't get to it. How do you put gas in your car. Lets stick with it and say you get to "cash out" an account of some kind. Guess what, they're not going to mail it to some random address. Now you're stuck fighting support centers to get them to understand that you need your funds delivered to New York even though you live in Florida.
In short, you don't need $100,000 in liquid funds, but there are a few cases that you need something liquid.
You also make a lot of assumptions. For example, not every one will have health insurance, or a heath problem that is covered by their insurance. Some serious home repairs are "down payment" upfront. Car problems like an accident that means you need a rental can totally be up front. Especially if credit cards can't be used.