Buying and paying off a house is one path to FIRE, but definitely not a given on the various paths to FIRE.
There are no rules.
It's a matter of figuring out what your priorities are: location, type of dwelling you need/want, commute, length of time you want to stay in one place, etc.
Rent, buy, house, condo, land, size, location, school districts, access to services, etc, etc, etc, these are all variables that you can scale up or down to find the right trade offs for your particular situation.
Don't try to follow "rules", that's how you fall into traps. Make real decisions based on real factors that are real FOR YOU.
Housing decisions are as big and as personal as career decisions. You cannot base them on simple metrics like: Mustachians do this or that.
Be smarter than that.