I keep almost all of my ‘emergency savings’ in index funds. Its value may fluctuate more than cash savings, but the higher return more than compensates. Unless, maybe, you face events that force you to tap it yearly or more, in which case I wouldn’t really consider it savings, so much as budgeting. I’ll generally have around a month-and-a-half worth of spending in cash, but that’s mostly because it’s going to get spent this month.
I think a significant part of recommending to keep cash savings falls to the legacy of financial markets. There used to be significant costs associated with transactions - I don’t mean $7 E-trade commissions, I mean $50+0.10/share - that meant it was really expensive if you needed to get $1000 out of the stock market for car repairs. There used to be significant lags: you’d call your broker, order the transaction, then have to wait several days for the proceeds to be delivered, a few more days for the check to arrive in the mail, then a few more days for the check to clear your bank; now, you order the trade online and they’ll have funds in your bank within 2 days at no cost. If you had an expense come up that needed paid today, then money locked away in the market was useless. Today, you put that expense on your credit card - which only became common in the 1970s - don’t need the actual cash for weeks, and can easily get investment funds that fast.
Example: I’ve been putting money into medical savings (USA) for a few years, which all goes into a S&P 500 index. That’s “down” about 10% from 2021, but I have around 40% net gains in the account. It turns out I have significant medical expense coming up, and it’s going to be a lot easier to deal with because of those gains. For scale, I’m expecting this thing to cost something like 3-4 month’s total spending and the healthcare savings is around 9 months.