The breakthrough can happen at any point in the future, tomorrow, or in 2 years, or 20 years. We don't know. What we know is that by allowing animal testing we most likely speed up that discovery process.
But we're not talking about experimenting on 15 million volunteer humans. We're talking about experimenting on 15 million animals incapable of consent. That figure should be equally morally repugnant to you.
Two can play at the hypothetical game. It's stupid. The same reasoning leads to moral acceptance of human organ harvesting. ("There's no willing donors, and those prisoners don't need an extra kidney anyway, and I do!")
(Though the concern that so-called animal "models" don't reliably predict the effect of treatments on human is very real.)
This is not the same as human organ harvesting, because that would result in death of humans, and we are trying to prevent that.
The concern that animal test results don't necessarily apply to humans is real, but it's much better than no tests at all.
A human and a fly? Sure, I see how one can apply vastly different moral judgements (I certainly do). A human and a pig? No, it's a matter of degrees.
And your hypothetical makes no sense. Discovering "a cure for cancer" takes much longer than the life expectancy of someone who is "dying from cancer".
Beside, my loved ones agree with me. My selfishness for companionship doesn't outweigh their moral stance.