You could use that logic to say keep all your savings in cash all the time. What if your roof gets blown off and you suddenly need to replace it? What if your house burns down and you need to buy a new one? The relevant question to ask is when having an extra couple of months' expenses in instant access is going to make the marginal difference. IMO the window between "one months' expenses" and "six months' expenses" is pretty narrow, so it makes more sense to keep one month or so in cash or current account and then invest the rest at the best rates you can.

(e.g. I would say for most people a car probably costs more than three months' expenses, so having that much extra in instant access probably doesn't shift you from "can't buy a new car in cash" to "can buy a new car in cash". Realistically you're (hopefully) insured and you're quite likely to finance your replacement car.)

I've had an "instant access" savings account take 2 months to pay out my money - some kind of AML issue, supposedly. Conversely, a lot of "three months' notice" accounts aren't "you physically can't take money out with less than three months' notice" but rather "if you withdraw with less than three months' notice you sacrifice one year's interest" or similar. I don't find the "hassle of an extra account" to be significant when it's two accounts at the same bank under the same login, but if you do then I would seriously consider having only a notice-requiring savings account.