'Rent is throwing money away' is mostly a myth
Renters pay for a roof over their head. Owners also pay a kind of 'rent', just to the bank (mortgage interest), the government (property tax), and the house itself (upkeep, insurance). You never get those costs back either. In the early years of a mortgage, most of each payment is interest, not equity. The honest comparison isn't 'rent vs mortgage payment'. It's 'rent vs the costs of owning you never get back, after you invest the deposit difference'.
How long you stay is everything
Buying carries large one-off costs: agent fees, stamp duty or closing costs, moving, and the cost of selling. Spread over 2 to 3 years these dominate. Spread over 10+ years they fade. If there's any real chance you'll move within about 5 years, renting is usually the lower-risk choice for your money. The longer you stay, the more buying's growing equity and fixed payment (which inflation makes cheaper) pull ahead.
Where inflation tips the scale toward buying
A fixed-rate mortgage payment stays flat in dollar terms while rents usually rise with inflation. Over many years that gap favours owning. Your housing cost is frozen while a renter's keeps climbing. This is the same effect the mortgage calculator shows as the 'real lifetime cost'. Fixed-rate debt quietly gets cheaper after inflation as the years pass.