Until you pay the mortgage. Then, your bigger expense, rent, disappear and you can start saving. Also, the thread of homelessness disappear from your life. That's something never will happen if you keep just renting.
that's not true. At any point in time, the cost of the capital is going to be comparable to rent. If you owned 100% if equity in the house, what you're "losing" is the potential rental income from the house, in exchange for the free rent you don't have to pay, plus the difference in the income from investing in the stock market (assume a maximally diversifed portfolio). AKA, imputed rental income.
The only bit you save is the tax that you didn't have to pay, if you actually earned the imputed rental income from your own property.
It's an illusion that living can be free. Renters pay in straight hard cash, so it's easy to count that cost. Owners pay in opportunity cost lost in the locked up capital in the house, or the interest payment (or some combination of interest and lost opportunity cost of capital).
I think that part of the problem is thinking of your home only in terms of an financial investment. A home is a place for living, you are always going to need a place for living. Despite appearances, you don't need a maximally diversified portfolio for living.