An emergency fund buys the right to have an emergency without immediately having another one.

Without a buffer, every problem becomes several. The car breaks, so the payment is late, so the fee is added, so the next month is short, and one event turns into a cascade. Savings do not prevent emergencies, they prevent an emergency from generating more of them, which is a different and more important function.

If you can put anything aside, even a small amount held for this purpose does disproportionate work. Keep it separate and slightly inconvenient to reach. And if you are currently in a cascade, the priority is stopping the chain rather than solving everything at once.