OTOH, In Reality, by the time I've managed to get enough money accumulated to pay my bills for the next six months, six months later, I usually find that I've used it up, , and was glad I could get to it. :-)
Read that again. It's saying that you should have six months of income in some form that is readily available. Cash under the mattress would be extreme, but an interest-earning savings account would be good. Then, anything beyond that six-month emergency fund should be put to work.
As you well note, reality often puts a chunk of that six month cash buffer to work defraying unplanned expenditures. Ask my transmission guy how I know. :0(