weekly or monthly salary payment is a good emergency that will make that person use his emergency fund to cover up until the person receives his salary. For instance, you used to receive your monthly payment at the end of each month, but something happened at the place of your work, and you got paid on the 10th of the new month. You will have no choice but to use your emergency fund to sort out your bills because you never expected that your salary payment would be delayed.
It seems you are confusing reserve funds for and emergency funds because that's actually not the work of emergency funds but however when we talk about emergency funds they are money kept aside that can not be expected to yield profits or use to solve daily needs however is mostly established for a purpose and for more crucial things, however when you talk about daily needs you are actually referring to reserve funds because those are the work of reserve funds they are some amounts of money that is removed from either monthly salary or other sources and save to help and sorts out any needs that would arise on the process of accumulation or holding.