On helping with the down payment, I would absolutely agree with you, especially if y'all are in agreement that you're buying it together. Now your other comment make more sense.

That's a completely different story than already having your own place and inviting someone to live with you. Maybe it's just me, but I would want my gf to pay a fair share of living expenses, even if I were managing it on my own or with a roommate (I'm married and we're stuck renting for now, so this was all hypothetical anyway).