The mortgage payment is always composed of two parts: the interest part, and the capital restitution part. When you pay a house with a mortgage, you payback your capital AND you pay interest on the capital which you have not yet paid back.
If, on the other hand, you just save and invest, you will earn interest on your capital instead of paying it. The "interest" you pay is the rent. Rent is nothing more, in nuce, than the interest on the capital you rent (the house). In a normal house market interest on houses is somewhere around 3 percent of the value of the house.
In a normal mortgage you pay much more than 3% on the capital you borrow. 5% is a much more normal rate.
The "down" payment doesn't change the logic. If, instead of buying a house now (let's say $300,000 down payment and $300,000 mortgage) you just invest the $300,000 you have and you rent a house of a value of $600,000 you will find that you will more easily buy a house after X years (your mortgage duration) by saving rather than by buying on credit.
As a banal example, the case may be that the $300,000 invested can give you 5% after taxes while the rent of the house will cost you 3% of the $600,000 house. That means the interest on the "down" payment will actually pay most of the rent. You will "save" the amount you would have spent for your mortgage and invest that. The "non-mortgage" minus the "rent not paid by active interest" is the actually "capital building" you make every year. You earn interest every year on that.
The idea that real estate increases with time is a prejudice. Real estate can increase, decrease, or remain stable. That's like saying that paintings increase value over time, or porcelain. There is no guarantee that houses rise in value in real terms in the long term.
The capital you invest will typically grow more than inflation for sheer economic reasons (we exclude particular, anomalous situations).
Don't buy. Rent, and save as if you had underwritten a mortgage.
Don't buy giving a "down" payment. Invest the "down" payment, rent, and save as if you had underwritten a mortgage (same logic applies).