People seem to use their mortgage offset account as an emergency fund. I’m not sure that LSL would work. A HISA would work similarly to a mortgage offset account, but why do you have one if it isn’t your emergency fund?
Firstly, a number of employers go down the gurgler owing employees LSL, annual leave and superannuation. So, if you’re planning to use your emergency fund on loss of employment, you’d be out of luck. There are provisions where the government picks up the tab, but it takes time.
Secondly, you seem to be assuming that you can cash out your LSL while you’re still working. Or are you only planning to use your emergency fund for when you can’t? There are plenty of situations where you will need an emergency fund and may still be working. What if your house burnt to the ground? You might not own it, but it takes a lot to replace your possessions and find somewhere new to live... Insurance only gives you so much, and you need to get your life back immediately. Certainly credit cards... are quite good at giving you a reasonable amount, but would that be enough in a timely manner?
So many people have not got income protection/TPD when problems occurred that I don’t trust that type of insurance at all.
I’ve had several emergencies in my life. Fire consuming every single possession except the clothes I was actually wearing; accident writing off my car and giving me whiplash for three years, causing a lot of time off work (and I fell through the cracks in insurance); job loss... You are often eligible for some compensation, but you may need emergency funds to cover extra, and to cover immediate requirements. I don’t think LSL would have worked in any of the emergencies I have faced.