You said that the purpose of an emergency fund can be defeated if a separate backup fund is not created for regular but non-urgent expenses like rent, school fees, gas refills. Therefore, it is important to have separate funds for real emergencies and regular expenses.
Rent, school fees, gas refills and general purchases are all part of your daily needs. So, if you want to invest, you have to invest with the remaining money (discretionary money) after all the expenses to meet these needs. This is mandatory for investment. You may never be able to survive in investment for a long time by investing outside your discretionary money.
However, in the case of backup fund, we should have such a strong plan that even in times of emergency, we do not have to dip our hands into the emergency fund. It is better for you to have at least 3 funds in the backup fund (1) Floating fund or cash flow (2) Reserve fund (3) Emergency fund.
When you are in an emergency, you may give priority to the emergency, so you will not have time to use the cash flow and reserve fund for any other purpose. At this time, you can try to overcome the emergency with cash flow initially, if it is not enough, then you can use the reserve fund. You can designate your emergency fund as a last resort, this may be the right time to use your emergency fund. Don't use your emergency fund in the early stages of an emergency, save it for the worst.