Surely the obvious point is that there are a lot of necessary parts, and there is no reason to pick out just one of them from the set and say the process "begins" with that one unless speaker means to ascribe uniqueness to it.
You're not wrong... but I think there's a bit more involved here. There are a whole lot of necessary elements for a pregnancy to occur. Just like with a car engine. Needs to have a central chamber, pistons, spark plugs, etc. Lots and lots of bits need to be there, all doing their part. That said... without gasoline, the engine cannot work. The gas is the catalyst injected into a system that makes it happen. If other parts of the system are lacking, sure, the engine won't work either. But the gas is still the catalyst.
Same concept here. There are a lot of elements - inside a woman's body - that have to be there for a pregnancy to occur. If there's not an egg, not a uterine lining, etc. there won't be a pregnancy. But the sperm is the catalyst.
And just like someone has to exert agency to put the gas in the tank to make the engine work... the sperm-injector is a voluntary actor that has complete control over whether the sperm gets inside or not.
A woman can say "No, I don't want to have sex.". She can say "No, I won't have sex without a condom" or "No, you cannot cum inside me". But at the end of the day, the woman cannot actually control whether or not the sperm gets there. No more so than the engine can reject the gasoline.