That's a lovely dream, I'd love to be able to trade/barter to get surgery.
It took me about four years of scrimping, selling, and a dedicated savings account to get enough money together, and I'll still barely make it. I'm going to be pretty strapped for a while but nothing else is this important to me. But I do recommend it, if you can either get a savings account that is specifically for your surgery fund, or get a special sub-account (some banks do this, not all do).
Just from my personal experience, when I had money in my regular accounts, I would spend it. If I immediately tucked some away in an account that was not connected to a spending account, it slowly accrued.