I'd be very curious to see the gender split on who considers contraception one of the most happiness-increasing inventions. My life would be very different without it, and I am so much happier with access to birth control, since it allows the (mostly) carefree sex I can have, and the fact that I don't have like 10 babies already.
I'm obviously a guy, but I don't think BC is a clear winner, or even a top contender, even considering the limitations you are putting in place. Eliminating drudgery has to be as important a factor in human happiness, and we have a ton of labor-saving inventions over the last 2 centuries. Washing machines, tractors, gas stoves, water pumps directly into the house...all sorts of really big things. I'd take 21st century of living with 10 kids over 16th century living with 2 kids.
Based on time use inventories, television and smart phones are both clear winners over BC, because they are clear winners over sex generally. People who are suddenly unemployed don't have a lot more sex, they watch a lot more TV.
There's also clear substitutes to BC, which is relevant to the final question "well, what's better than happiness?" A world without BC is a world with more abortion, probable infanticide, and where women are stuck in marriages with men. This is obviously inferior to the current state of affairs, but that's not a zero-happiness world, it's a less-happiness world.
Also, people prior to BC still had sex, even though they were impoverished relative to us. I wouldn't say it was "care-free" sex, but they for the most part thought the risk was clearly worth it.