I agree $1000 is not a lot for some people

Geographic location and number of people in your household matters too. If a single person has $1,000. It can go much farther than a family of 6. That is because there can be more “unexpected expenses” when you have other people’s health and vehicles to attend to.
Our DD just got her cell phone stolen off the bus a couple of days ago. The deductible was $150. I took that from our BEF because I’m scrapping my budget now to complete my Christmas shopping and didn’t want to take that from our working capital. After Christmas is over, I can budget in my January budget to put that $150 back. In the past, I would have just charged it on my credit card. Since I don’t have those anymore, I use my BEF for that purpose.
Other emergencies like car repairs, household issues like broken appliances, and health deductibles could be drawn from a sinking fund. We know we have a $3,600 deductible for health issues, we put money into our HSA for that purpose. We do not draw that out of our BEF. Same thing with car. I put money every month into a sinking fund for that. Appliances could be another category especially if you know your water heater, frig, or dishwasher are 18 years old!
It’s all about planning for those things. I rarely use the BEF.